Gravitational and inertial mass seem to result from different fields, yet virtually the same. How do new physics theories attempt to reconcile this? As best I understand it, internal mass (at rest) is due to interactions of particles with the Higgs field, and in equation terms, due to various chiral terms, sometimes described as swapping chirality.
By contrast, gravitatjonal mass (at rest) would be due to interactions with a completely different field, or structure (unless within some theory that hypothesises that the Higgs field is also responsible for gravity as well).
And yet, within fairly narrow limits of experimental.error, they appear the same.
What ways do theories of "new physics" approach this, in order to end up with the two values being virtually the same within them?
 A: the most clarifying information on this topic that I am aware of is in a physics blog post (2007) by a physicist called Jim Pivarski, titlled The origin of mass?.
(I noticed your 2018 question about the unification of electromagnetism and the the weak nuclear interaction. So I'm aware of what you already know, but I will write a standalone answer here.)

The introduction of special relativity forced a fundamental rethinking of the concept of inertial mass.
Einstein offered a thought experiment leading to the implication that in terms of relativistic physics it is necessary to attribute an inertia to the energy of light, with the amount of inertial mass being expressed with the relation $m = E/c^2 $. Moreover, that implication has general validity. If special relativity is correct then a logical implication is that a spinning flywheel has a larger inertial mass than it has in non-spinning state.
(Of course, macroscopic flywheels cannot reach the kind of angular velocity that would give rise to a measurable relativistic effect. A macroscopic flywheel will shatter way before reaching such a velocity.)
Jim Pivarsk writes:
"[...]the protons and neutrons are known to be made of quarks, bound by an incredibly strong force called The Strong Force. Converted into conventional units, quarks attract each other with forces typically greater than 15 tons. The potential and kinetic energy of the quark orbits account for 99% of the mass of protons and neutrons; only the last 1% is due to the mass of the quarks themselves. This is relativistic mass in an extreme case [...]"
The point is: this inertial-mass-arising-from-kinetic-energy is outside the scope of the Higgs mechanism.

Electroweak interaction
The theory of electroweak interaction offers a way of unifying the descriptions of electromagnetism and the weak nuclear interaction. This requires a mechanism to account for the fact that while the electromagnetic interaction is a long range interaction the weak nuclear interaction is inherently short range. This can be accounted for if the particle corresponding to the weak field is not massless, but instead has a significant inertial mass. Moreover, given the range of the weak nuclear interaction it could be inferred what that mass would have to be.
Pivarski writes that the Higgs mechanism is as far as known the only way to make the weak nuclear interaction short range.

On the subject of the relation between the Higgs mechanism and inertial mass Jim Pivarski prefers the following interpretation:
"[...] the electron-Higgs interaction, with the background Higgs field taken as a given fact, can be expressed as a quadratic potential for electron field. This potential provides an energy cost to exciting the electron field from a zero-electron state to a one-electron state. [...]"
That is, as I understand it the way the Higgs mechanism introduces inertial mass is by imposing an energy cost, and the resulting inertial mass can then be interpreted as the inertial mass of that energy.

Neutrino mass
According to the current models that account for the existence of neutrinos the neutrino's must have a non-zero rest mass. This rest mass cannot be accounted for in terms of the Higgs mechanism.
The exact value of rest mass of neutrino's is unknown. Over the years larger and larger experiments, including dedicated experiments, have lowered the upper bound. (Large experiment that is expected to give results soon: KATRIN)


My takeaway from the above is that the Higgs mechanism does not provide an exhaustive account for what inertial mass is. There are aspects of inertial mass that are outside the scope of the Higgs mechanism.
So I think the premise of your question isn't necessarily the case: inertial mass and gravitational mass do not necessarily arise from different fields.


We have that in terms of GR inertial mass and gravitational mass are, as a matter of principle, the same entity.
For now let me refer to that entity as 'inertio-gravitational mass'.
(John Stachel, director of the center of Einstein Studies, has proposed the name 'inertio-gravitational field' for the field that is described with the Einstein Field Equations. This proposal dates from the 1990's)
To my knowledge there is currently no attempt at formulating a theory of inertio-gravitational mass. In order to formulate a theory at all the properties of inertio-gravitational mass must be granted.
The equations of GR include various contributions, these contributions are chosen such that the theory satisfies the constraint of equivalence of inertial and gravitational mass. That is: the particular form of the contributions that go into the equations is a consequence of taking the principle of equivalence as starting point.

It may be that in a future theory the equivalence of inertial and gravitational mass will be abandoned. In such a theory the relation between inertial mass and gravitational mass will be accommodated by way of adjusting one or more adjustable parameters.
Current attempts, to my understanding, assume the properties of inertia as is, in order to be formulated.



The above is towards answering the question, but there is a general issue that I think is worthwhile writing some more about:
There is a fundamental distinction between explaining an entity/property in a theory, and granting that entity/property in order to formulate the theory.
Example: compare the thermodynamics of Carnot and Clausius to the statistical mechanics of Maxwell and Boltzmann. Carnot granted certain physical relations between energy and heat, and from there developed a fruitful theory of thermodynamics. Statistical mechanics, formulated in terms of motion of atoms and molecules, addresses the physics taking place at a deeper level of description.
In Carnot/Clausius thermodynamics entropy is accomodated as is. Statistical mechanics explains the nature of entropy in a way that is inaccessible in terms of Carnot/Clausius thermodynamics.
Of course, statistical mechanics is still granting things in order to be formulated at all, but those are at a deeper level of description.
In terms of GR: in order to formulate the theory at all the equivalence of inertial mass and gravitational mass is accommodated as is.
