How are probabilities "physically real"? I see several related questions, mostly taking the standard position/interpretation that complex-valued amplitudes aren't "physically real" (though their imaginary components do participate in interference effects that are ultimately experimentally observable), whereas real-valued probabilities are "physically real". And I'd agree with that.
However, although probabilities are observable, they aren't canonical operators in the usual Hilbert space or $C^\ast$-algebra sense. You "observe" probabilities by repetitions of some relevant measurement, and then applying the standard Kolmogorov frequentist interpretation of probability to the set of resulting outcomes.
So that's hardly a Hermitian operator, or anything axiomatically identified with typical "observables". It's pretty much just the Born rule. So if we'd agree that bona fide observables are "physically real", then calling probabilities  "physically real" must be using that term in some different sense. So what are these two distinct senses of "physically real"; in particular, what distinguishes the "physical reality" of probabilities from that of typical observables?
Edit Note that the Born-rule probability calculation is foundationally different than, say, the calculation of an apple's volume from a measurement of its radius, which we might call the sphere-rule. You can design an experimental apparatus that directly and independently measures an apple's volume (though I'd refrain from running naked through the streets crying "Eureka!" about it:), whereas there's no possible way for an apparatus to >>directly<< measure probabilities at one fell swoop, so to speak.
I mention this in response to Ion-Sme's answer: Since there are direct measurements of gravitational fields, that wouldn't be an apt analogy relevant to the purpose at hand.
Edit The comments from @Nathaniel and answers from @BobBee and @StéphaneRollandin seem to all conclude (in Stéphane's words) that it "depends on what you mean by 'real'". That kind of seems to suggest there's no definitive answer, and even the definition-dependent "answers" here are basically discussions, i.e., nothing formal and rigorous. So let me try (without completely succeeding) to make the question a little more definite, as follows...
In the current (March 2018) issue of APS News, there's a page 4 obit for Polchinski, where they quote from his 2017 arXiv memoir, in part, "I have not achieved ... explaining why there is something rather than nothing." So can we agree with his conclusion that there's something rather than nothing? Okay, then in that case, I'd suggest that those non-nothing somethings are, by our definition here, "real".
I realize that's still pretty wishy-washy, almost certainly insufficient for any subsequent rigorous treatment stemming from those words. And that's kind of the whole point. We've got Polchinski saying, "there's something rather than nothing", and he presumably wasn't wishy-washy philosophizing. Instead, he was lamenting over not achieving that ambition of explaining why there's this "something". But if he'd wanted to rigorously mathematically achieve that explanation, it would prerequisitely require some rigorous axiomatic-worthy definition of the "something" you're going to subsequently explain.
So this doesn't really define "something"~"real", but I think it's a reasonable argument why such a formal definition is necessary, and physics-worthy rather than just philosophy-worthy. Physics formally, rigorously, mathematically discusses the observable "somethings" of experience. But then it can't say what those "somethings" are??? Seems like that's an important missing piece of the puzzle.
 A: You've made your point, both pro and con the reality of quantum mechanical (QM) probabilities. Is the question about physics or about interpretations or semantics or as some experimentalists would say, philosophy?
Please note that in some of the various interpretations (not theories) of QM one can hear that the position of particles, a QM observable, is not real until it is measured. So the semantics or interpretations issues go all the way to the basis of QM. Is the wavefunction real, a real thing that IS the particle, or is it a mathematical tool to track the particle, and only when you measure (or the environment does by collapsing the wavefucntion through interactions) it do you see any reality? 
The proBability (the absolute square of the wavefunction) is as real as the phase, one determines interference, but the other one determines likelihood, in QM. Really, the phase only manifests in the probability of the two summed, interfering, wavefunctions. You can't have a phase unless you have a wave, i.e. a wavefunction, and thus the probability. 
So, for both phase and probability, i.e. amplitude, it's the same issue. 
The QM interpretations and the Bell Theorem (which simplistically says that naive reality and locality cannot both be true, see the first reference below) have more to say on all this, but some physicists (most?) would say that that makes no difference, you can still use QM to calculate results and measure things to confirm or deny. I refer you to a couple of references, the first one on Bell's theorem which talks about what reality is assumed to mean and altErnatives, the second on QM interpretations. 
https://en.m.wikipedia.org/wiki/Bell%27s_theorem
And 
https://en.m.wikipedia.org/wiki/Interpretations_of_quantum_mechanics
A: As @BobBee rightly points out, the point you make goes down to the whole mathematical structure of QM, and depends on what you mean by "real".
Let's take "real" as labelling the intuitive meaning we give to objects and their properties in classical physics. Such objects and properties are: objective, well-defined, and thus associable to a matter of fact. You can lend them to your neighbor, and he can give them back to you. You can forget them in the sand, they will still exist. They have some "state".
In this regard a QM state has a very misleading name: it definitely does not describes something real as defined above. It describes an object inasmuch as we trust that talking about this object will keep some sense throughout, because if you look in the QM state itself with a classical mindset, you find a superposition of classical descriptions weighted by probability amplitudes - there is a shift here from describing an object classically to describing the object as a non-trivial structure mixing several classical descriptions of the, well, object. Something is going on here: we do not speak of the same thing, we are not at the same level of representation. The QM description is a meta description: a description about descriptions. 
And the normalization has to be done by hand, so that Hilbert rays form an equivalence class. This is very fishy: unitarity is fundamental, else probabilities do not hold, yet we have to normalize by hand. This shows again that it is rather explicit in the formalism what we know all along that we are talking about representations, possible representations, interfering representations, not about the object or physical system itself. The classical description of the object is still there, but it is in the Hamiltonian. In the Hilbert space, what you have is not a description of a real (again in the sense defined above) object, but a description of how it manifests itself when some measurement occurs. 
The dreaded word appeared so I have to stop here: measurement. This is the measurement problem, again. Why can we not speak about an object, but only about measurement of that object, even while "measurement" is the most ill-defined theoretical notion in QM?
A: "there's no possible way for an apparatus to >>directly<< measure probabilities at one fell swoop"
But you can approximate them by repeating a measurement, and counting the relative frequencies of the different outcomes. 
A: Neither QM nor QFT is allowed to change the definition of a probability without creating a new term; otherwise, it would be patently irresponsible to use the term in order to verbalize theoretical mathematical conclusions.
Our difficulty comes when we are uncertain of a future occurrence without destroying the evidence of its correctness.  We are forced, then, to assume the existence of every possible outcome in order to move forward.  Fundamentally, however, regardless of the probability of an event occurring, once it does occur, in that instant, the probability is one.
Probability distribution parametrics are real only if they are defined to be so.  Expected values are not defined to be real (2.7 average family size); however, a median value is a real possible outcome. Discrete distributions are also real possible values. We can also create a confidence interval that states with some level of certainty the actual location of a particle. But, with a confidence interval we are only back to our original problem of assuming all probable values within that interval are real in order to proceed.  Therefore, given that we must bring forward in our calculations the pseudo existence of all location possibilities, our final solution must necessarily create the same number of solutions. Even if we insist our probability distribution is actually discrete in order to ensure real answers, the best answer we can get without measurement is a probability of a real answer within a range of values.
Our good fortune is that it doesn't matter precisely which is the real solution because of the computation collapse of all the possible solutions to the same number.  Thus the apparent magic of QM & QFT.
For over a century, this magic has made many unnecessarily uncomfortable  with the representational implications of the answers. "Shut up and compute" remains the watchword of about a third of the practicing physicists.  Unless one can demonstrate a violation of the mathematical equivalence needed in our translations as we move into Hilbert Space, there is no reason yet, other than discomfort, to doubt the representational nature of these calculations. Thus the principles of Effective Field Theory remain valid with all of its profound implications.
