Famaey & McGaugh's review <https://arxiv.org/abs/1112.3960> (Stacey McGaugh is <https://tritonstation.com/>, a frequent critic of particle dark matter) §7 is all about adding whole new fields ("parameters") to standard General Relativity in order to capture higher-order terms in post-Newtonian expansions (in 1/c^n or in the metric tensor) of theories that preserve the central characteristics of Milgrom's MOND.

Those characteristics are: no "non-luminous" or "hidden" matter sources -- the glowing, refracting, and obscuring dusts are the sole sources of gravitation -- and in circumstances where post-Newtonian corrections are vanishingly small, strong concordance with Newton and Kepler except at the lowest accelerations.

The orbits of galactic blobs of slowly-moving molecular gas that produce characteristic spectral lines amenable to Doppler redshift study are almost always in those special circumstances, and they have higher accelerations "corewards" in galaxies and low accelerations "outwards" in galaxies. These gas blobs are most interesting at the leading and trailing limbs of edge-on discoid/spiral galaxies, or face-on in large elliptical galaxies with negligible bulk rotation (the Doppler-shifting blobs move in and out, almost entirely radially), precisely where the accelerations are smallest.

The question we then ask is: what if we have a blob or some other spectrally well-defined object in a galaxy is on a very fast orbit? We can no longer ignore post-Newtonian corrections, or we lose the MONDian match with spectral lines. This is what drives §7 ibid.

If we think of the expansion as:

Total gravitation = empty flat background + MONDian matter relation + higher-order terms (h.-o.t.s. or HOTs)

the relativistically moving but still-MONDian matter is the generator of the HOTs, which has been measured (by among other people Pavel Kroupa, the author of the fine article at the top of all this discussion).

Abandoning the MONDian matter relation is easy. Just recast:

Total Einstein gravitation = empty flat background + unknown stress-energy tensor + higher-order terms

and adapt the stress-energy tensor. Nothing in the theory of gravitation has ever suggested strongly the nature of matter which generates the stress-energy, but we can get constraints from other areas of physics like the standard model of particle physics (or even classical electromagnetism, as was the case when General Relativity was new), and we can get candidates from programs seeking to expand or alter the Standard Model to solve open non-gravitational problems. Thus: various types of electrically-neutral particles like axions or supersymmetrical sparticles have been considered because "someone else" (i.e., not a relativist) hypothesized them. (Observational cosmologists can find evidence to constrain such theories, and in practice helped kill off several supersymmetry extensions of the standard model, for example). Low-mass primordial black holes were a candidate too, because there were non-galactic-dynamics reasons to suspect they could exist in significant number. But those black holes (or very dim tiny stars or isolated cold Jupiters) could maybe generate the unknown stress-energy tensor.

Retaining the MONDian approach can be done in a couple of ways. Hold to the idea that the "luminous" stuff is the only source of gravitation, and then ask whether relativistically-moving MONDian matter is coupled to an unexpected background:

Total gravitation = empty flat background + curvature corrections + MONDian matter + HOTs

(where all that's left in HOTs is basically gravitational waves from relativistically-moving MONDian matter)

or alternatively whether there is a further field or fields to which slow-moving MONDian matter couples only very weakly, but fast-moving MONDian matter experiences quite strongly. That means the HOTs imply the existence of one or more new fields.

Are these fields free parameters? Good (and open) question.

What types of fields are allowed? §7 ibid. explores this largely in terms of adding fields to the Einstein Field Equations (EFEs) of general relativity. The fields conceptually modify Einsteinian gravitation, so are written on the curvature side of the EFEs. However, as the authors of the survey note, in most cases for purely mathematical reasons the fields could be written exactly equivalently as a function on the stress-energy tensor (the matter side of the curvature = matter relation).

Why are we thinking in terms of an adaptation of the Einstein Field equation in the first place? Because it works well for length scales much shorter than ~ten kiloparsecs, and might work well for length scales much greater than megaparsecs. In particular, it works so well in the solar system that it cannot be ignored: it is the effective theory of gravitation near us. So however one might eventually write down a relativistic MOND theory, it must be possible to re-write it into a modification of the Einstein Field Equations for use in the solar system and in systems like Hulse-Taylor or triple-pulsar J0337+1715.

Additionally, analysis through tools like the Parameterized post-Newtonian formalism <https://en.wikipedia.org/wiki/Alternatives_to_general_relati...> is extremely useful if one recasts some arbitrarily-different-from-General-Relativity gravitational theory into something for which PPNF variables can be found. The discipline of submitting to PPNF by rewriting a theory for PPNF friendliness often is personally useful for the theorist keenly interested in her or his not-like-General-Relativity theory of gravitation.

One might say, aiming for neutrality, that if one adapts the EFEs in such a way that the adaptation can appear on either side of the EFEs, the "conceptual" weight of the adaptation is perhaps more aesthetic than physical.

However, that's not quite fair: outright adding extra stress-energy dark-matter-style lets one decouple the "adaptation" of the galactic relation from galaxies themselves. One could put particle dark matter in parts of the universe where there are no electrons or protons. For instance, before electroweak decoupling and big bang baryogenesis, or in the extreme distant future where we have basically isolated black holes and relic microwaves (and neutrinos) that are undetectably cold/long-wavelength. Even around "today"'s epoch, dark matter could be put willy nilly well away from clusters of galaxies.

A modified gravity theory, relativized along the lines of Famaey and McGaugh's surveyed options, might also be able to do this. Strange curvature just existing apart from any matter. (One can do this by a choice of a strange background in a post-Newtonian correction formulation of standard General Relativity too.) But to be MONDian I think that you would want attractive gravitation to appear only where there is matter that interacts electromagnetically, because far from such matter (attractive) accelerations will be very low, and the central feature of MOND is that Newtonian gravitation needs adjustment at very low accelerations.

So for a fully-relativistic MONDian theory of gravitation, although maybe mathematically we could treat the low-acceleration as a field that we could drop just about anywhere, doing so would violate MOND's spirit. There is thus no vacuum + MOND field(s) gravitational solution that is remotely physical. This is certainly not the case for particle theories of dark matter. Those basically insist that it is perfectly reasonable to drop dark matter just about anywhere. One can easily expect to simulate a vacuum + cold dark matter solution to the Einstein Field Equations, and that the simulation might accord with some part of our actual universe.

Finally, without further diving into the aesthetics or bets about what particle physics at CERN and the like might discover about mechanisms that generate stress-energy, I think the "fight" between MONDian modified gravitation and Einsteinian gravitation with dark matter is irrelevant to astrophysicists (as opposed to theorists who deal with non-astrophysical systems), even those who work on galaxy dynamics. It is almost certain that it will be possible to find an initial values formulation for either type of "final conceptual" answer to the observed Milgrom relation. It's just a question then of finding out how to populate the initial values surface, and then letting the dynamical laws go to work on those.

(The initial data and laws for each theory (MOND or GR+DM) reformulated this way will necessarily differ, perhaps by a lot, but the approach of evolving a set of initial data will be the same (this is done in many areas of physics having little/nothing to do with gravitation, after all)).

That is, the core of the MOND/particle DM fight might be more about bets on whether the single minimally-coupled metric tensor approach in General Relativity is not always suitable, or whether the Standard Model of particle physics is incomplete in relevant ways, than about whether one is really more suitable for calculations than the other. (The bet is also not strictly either/or!)

I'm honestly not sure who will really care in practice, if it's ever "finally" decided one way or another.