What is the experimental evidence for the gravitational field having positive energy density? Recent direct observation of gravitational perturbations attributed to merging black holes and merging neutron stars has reliably confirmed the existence of gravitational waves. The observed fact that the orbits of binary pulsars decay, and the assumption that the decay is due to emission of gravitational waves, is interpreted by some to imply that the energy density of the gravitational field is positive.
Here're relevant quotes from Steve Carlip:

*

*

To make gravity attractive in such a [vector-like] theory, you must require that the gravitational field has negative energy, which (apart from the obvious instabilities) would drastically disagree with binary pulsar observations



*

In electromagnetism, opposite charges attract, and electromagnetic waves carry positive energy.  To get like "charges" to attract, you have to change a sign, and this changes the sign of the energy carried by a wave.  In a vector theory of gravity, where like masses attract, a gravitational wave would have to carry off negative energy.  This would mean that a binary pulsar system would increase in energy as it emitted gravitational waves, which contradicts observation.

Observations do seem to show that the energy content of the gravitational waves emitted by the orbiting pulsars is positive, but it's not obvious (to me, at least) that this requires the energy density of the gravitational field to be positive. For example, suppose the gravitational field energy density is negative, and a gravitational wave is a perturbation in which the field strength is reduced within the wave.  This should give the gravitational wave net positive energy, but allow the gravitational field energy density to be negative.
This article by Guth at Cal Tech appears to contradict Carlip:

The resolution to the energy paradox lies in the subtle behavior of gravity. Although it has not been widely appreciated, Newtonian physics unambiguously implies that the energy of a gravitational field is always negative a fact which holds also in general relativity. The Newtonian argument closely parallels the derivation of the energy density of an electrostatic field, except that the answer has the opposite sign because the force law has the opposite sign: two positive masses attract, while two positive charges repel. The possibility that the negative energy of gravity balance the positive energy for the matter of the Universe was suggested as early as 1932 by Richard Tolman, although a viable mechanism for the energy transfer was not known.

The widely used WEC (weak energy condition) asserts that the energy density everywhere is non-negative in GR; but in most of the references I've read, the context is cosmological, and there is an assumption that the universe is filled with a perfect fluid having a mass density.
Questions:

*

*Is there other observational evidence that can clarify this?


*Are there inescapable theoretical arguments in General Relativity that require the gravitational field to have negative energy density?

Edit (2018-05-28)
Patrick Dürr argues persuasively that gravitational waves do not carry energy.
Another perspective on energy density in the gravitational field and in gravitational waves (the two not being exactly the same!) is given by the last contribution by "Demystifier" in Physics Forums.  There, it is argued that although the energy density of the gravitational field is negative, the perturbation due to a gravitational wave essentially reduces local gravitational field strength, so that the wave thereby effectively carries positive energy despite the energy density of the gravitational field being negative. This, if correct, might help resolve the otherwise apparently conflicting positions of different authorities on the subject.

Edit (2018-06-28)
Another set of possible observational evidence would be direct measurements of the Sun's gravitational field as a function of distance from the Sun. If the static gravitational field's energy density is positive (or negative), and energy is a contributing source to the gravitational field, then the gravitational field will not vary exactly as $1/r^2$; and the deviation from $1/r^2$ should tell us something.
Deviations near a black hole or neutron star would probably be easier to measure.  However, the unknown distribution of dark matter is likely to make the interpretation of such measurements very uncertain.
 A: I have dug out as many papers as I could find, dealing with the question of whether the energy density of the gravitational field is positive or negative.  As it turns out, the authors fall into at least three different camps:


*

*Energy density is negative.

*Energy density is not really definable

*Energy density is positive.


In the first camp are those who think in classical Newtonian and Maxwellian terms, representing gravitation as a vector field directly analogous to the electromagnetic field.  This kind of field, in order to provide an attractive force between masses, requires the gravitational field to have a negative energy density.  In itself, this doesn't seem to be a problem – until gravitational radiation is considered.  With negative energy density, gravitational radiation emitted by rapidly oscillating masses (e.g., two neutron stars in close orbit) is expected to carry negative energy.  As a result, the masses would be expected to  gain energy.  That is, a pair of orbiting neutron stars should get farther and farther apart as they emit gravitational waves-- which is contradicted by observation.
In the second camp are those who understand general relativity well enough to know that it is mathematically impossible even to define an energy density that is consistent with the principles of GR.  In this camp,  local field energy density is considered a fiction, only total energy is conserved, and total energy is assumed to be positive (this is the “Weak Energy Condition”).
In the third camp are those who understand the problems with #1 but either haven't bought into #2, or are actually members of #2 but over-simplify their explanations in hopes of communicating with an under-educated audience.  Also in the third camp are a substantial number of researchers who examine what happens in the weak field limit, where the nonlinearities of gravity in GR become inconsequential.  That boundary between camps #1 and #2 seems to be a somewhat dangerous terrain with positive energy density on one side and negative energy density on the other side.  There is no obvious conceptual path by which a theoretician can travel smoothly and safely from one side to the other – unless the total energy of the universe is assumed to be zero, in which case an energy density apparently can be defined. 
I started out from a  Camp #1 perspective.  Steve Carlip figuratively gave me a sharp thump on the head, which led to my question that started this conversation.  Without a much deeper understanding of GR I can't be in Camp #2.  I'm searching for that smooth, safe path from negative energy density to positive energy density, in a weak-field approximation to general relativity.
At this point I'm ready to offer my own answer to my 2-part question.  
First, observational evidence that the gravitational field has (effectively) a positive energy density includes a) observations that binary pulsar orbits lose energy over time, and b) direct observation of gravitational waves that “chirp”, indicating rapid loss of energy in the moments before black holes or neutron stars merge.  Second, although there are strong arguments against the gravitational field having a negative energy density even in the weak-field limit, the arguments have not yet reached the point where they can be called inescapable. Some covariant scalar-tensor theories provide positive energy densities, and some versions of Einstein's equations provide terms that give a net positive (quasi) energy density for the whole field despite having a negative energy density for the part attributable to the gravitational field per se. There is plenty of theoretical work yet to be done relating to questions about gravitational field energy density.
That would seem to wrap it up, but the answer certainly doesn't have a feel of completeness.  For example, it's not certain that all possibilities have been found and considered in the search for vector-like covariant gravitational theories whose field equations imply negative field energy density but describe  waves that carry positive energy. 
The points mentioned above are based on what I've gleaned from a large number of papers, but the following papers are particularly relevant: 
John D. Norton    EINSTEIN, NORDSTRÖM AND THE EARLY DEMISE OF SCALAR, LORENTZ COVARIANT THEORIES OF GRAVITATION  here (An excellent and very readable analysis of papers presented by, and communications among, key players in the evolution of relativistic theories of gravitation.)
Cheng Zhang et.al.  Energy Density of Gravitational Field  in General Transverse Gauge  here (describes a linear weak-field approximation to Einstein's equations, with positive field energy density.)
Diogo Bragança   Energy in general relativity: a comparison between quasilocal deﬁnitions here (shows how different interpretations result in positive or negative energy densities.)
Ed Witten    A New Proof of the Positive Energy Theorem here  (offers a proof that the total energy of the universe is positive.)
Neil Dewar  On Gravitational Energy in Newtonian Theories here (shows that the concept of field energy density has subtle complications even in Newtonian gravitation.)
Thibault Damour1 1974   The discovery of the ﬁrst binary pulsar here
Yu. Baryshev     Energy-Momentum of the Gravitational Field:  Crucial Point for Gravitation Physics and Cosmology  here (offers a quick review of the problem of energy density in gravitational field theories and develops a positive energy density scalar-tensor theory of gravity.)
A.I.Nikishov ON ENERGY-MOMENTUM TENSORS OF GRAVITATIONAL FIELD here  (derives several different expressions for a gravitational field energy density; discusses issues)
A: In Newtonian gravity, the energy density of the gravitational field is negative. This is necessary in order to explain the attractive nature of the gravitational force. The energy density is $\propto g^2$, where $g$ is the gravitational field and the constant of proportionality is negative.
In general relativity, the equivalence principle guarantees that the gravitational field cannot be a well-defined thing. In a free-falling frame, at a given point, the gravitational field is always zero. Since the gravitational field $g$ is not well defined, GR cannot have an energy density $\propto g^2$.
What GR does allow us to calculate is the energy content of a gravitational wave, averaged over some region of space whose size is $\gtrsim$ the wavelength. The energy of such a wave is positive. For more on this topic, see https://physics.stackexchange.com/a/402942/4552 . It is not controversial whether gravitational waves carry energy (and it has not been controversial since the 1950's). It is not controversial that their energy is positive. (See below for experimental evidence.) There are expressions that we can integrate in order to find the energy of a gravitational wave (see the link above), but those cannot be interpreted as energy densities that exist at every point in space. They are only of use when integrated over a region large compared to a wavelength.

What is the experimental evidence for the gravitational field having positive energy density?

The gravitational field does not have a positive energy density, because the gravitational field does not have a well-defined energy density. A gravitational wave does have an energy, and it's positive, but the energy cannot be localized to $\lesssim$ the wavelength.
As far as experimental evidence, the Hulse-Taylor system is seen to lose mechanical energy over time. Since the ADM energy is conserved in an asymptotically flat spacetime, this requires that the gravitational waves emitted by the system have an energy that is nonzero, positive, and has a definite value, which we know. The results from this system are in excellent agreement with theory.
The OP seems to believe that there is controversy on this issue among experts. There is not. The quotes from experts like Carlip are being misinterpreted and taken out of context.
Here are a couple of references that discuss this sort of thing:
Wald, General Relativity, 1984, p. 85
Misner, Thorne, and Wheeler, Gravitation, p. 964
A: 
What is the experimental evidence for the gravitational field having positive energy density?

Clocks go slower when they're lower, all the more so when the gravitating body is more massive. 

Recent direct observation of gravitational perturbations attributed to merging black holes and merging neutron stars has reliably confirmed the existence of gravitational waves. 

I'm not sure how reliable they are, but I don't have any issue with the existence of gravitational waves.  

I understand that the observed fact that the orbits of binary pulsars decay, and the assumption that the decay is due to emission of gravitational waves, has led to a consensus that the energy density of the gravitational field is positive. 

I don't think there is a consensus on that. I'm forever saying gravitational field energy is positive. I point to page 185 of the Doc30 Foundation of the General Theory of Relativity and Einstein saying “the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy”. And yet I find there are people who object, and point to physicists and cosmologists who claim the opposite. For example Lawrence Krauss claims gravitational field energy is negative, and that the total energy of the universe is zero.  

Here's a relevant quote from Steve Carlip: "To make gravity attractive in such a [vector-like] theory, you must require that the gravitational field has negative energy, which (apart from the obvious instabilities) would drastically disagree with binary pulsar observations".

I'm not clear on the context there. Gravitational attraction works in a particular way, and it doesn't depend on the spatial energy density, it depends on the gradient of the spatial energ density. If there is no gradient, there is no gravity.   

It does seem to imply that the energy content of the gravitational waves emitted by the orbiting pulsars is positive, but it's not obvious that this requires the energy density of the gravitational field to be positive.

IMHO it's obvious because a field is a standing wave, or a wave is a dynamical field-variation propagating through space. 

For example, suppose the gravitational field energy density is negative, and a gravitational wave is a perturbation in which the field strength is reduced within the wave.

The energy density relates to potential, whilst field strength relates to the gradient in potential. The latter causes you pencil to fall down. It falls towards the place where energy density is greater. If the wave is very large such that there's no discernible local gradient, you might be able to detect distant pulsars speeding up whilst you and your clocks are subjected to gravitational time dilation.     

It seems that this would give the gravitational wave net positive energy, but allow the gravitational field energy density to be negative.

I don't see that I'm afraid. 

Is there other experimental evidence that can clarify this? 

Yes, gravitational redshift. See what Einstein said: "an atom absorbs or emits light at a frequency which is dependent on the potential of the gravitational field in which it is situated". When the E=hf photon ascends, its frequency doesn't actually reduce. Conservation of energy applies*. It appears to have less energy at the higher elevation because that's where the spatial energy density is lower, so we and our clocks go faster. So we measure the photon frequency as reduced.  

Are there inescapable theoretical arguments against the gravitational field having negative energy density?

IMHO general relativity is fairly inescapable, as is classical electromagnetism, and they both say field energy is positive. I don't know of any theory per se that says field energy is negative.  


*

*Some say conservation of energy does not apply in general relativity. I do not agree with that.
A: General Relativity in an expanding universe does not have a conservation law for energy, so a typical balance of energy can only be applied in a flat slice of spacetime
If you have masses $m$ and $M$ with rest mass $(m+M)c^2$ and gravitational potential energy $-G \frac{mM}{R}$, this energy would reach zero if 
$$ R = \frac{G mM}{(m+M)c^2} $$
which is about the order of the Schwarzschild radius of the gravitating system
One can interpret this in a number of ways, but my preferred interpretation is that the Newtonian definition of gravitational energy fails right before allowing the total system energy to become negative.
