Interacting QFT construction on curved spacetime As far as I can tell, most of the concrete models considered in (rigorous) QFT on curved spacetime are either free or perturbative. In fact the only construction of an interacting QFT on curved spacetime that I am aware of is by Figari, Höegh-Krohn and Nappi (PDF), developed further by Barata, Jäkel and Mund, for $P(\phi)_2$ theories on de Sitter space. This work uses the Euclidean approach via a novel kind of OS reconstruction.
In generic globally hyperbolic spacetimes, Euclidean methods no longer work. Given their usefulness, this is quite a blow to the constructive programme and I assume some new ideas will be needed. (There's a very interesting article about the unitary (un)implementability of dynamics for QFT on curved spacetime that indicates how some very basic concepts seem to break down for such theories.) There is some recent AQFT work on constructing the C*-algebras of interacting fields, I am also vaguely aware that Wald and Hollands suggested using OPEs, but it is not clear to me how successful these ideas have been or will be in producing actual models.
My question then is for those more in the know: are there at present any ideas in the community for constructing say $\phi^4$ theory on a general globally hyperbolic $2D$ spacetime, i.e. when Euclidean methods break down (besides the C* and OPE frameworks already mentioned)?
EDIT: Here is a reason to expect such a thing to be possible at least sometimes. The main problems in constructive QFT come from controlling the UV and IR behaviour of the theory. On the one hand, the UV behaviour of a theory on curved spacetime should really be very similar to flat spacetime since every space looks locally flat at small scales (perhaps if we forget about singularities for the moment), while we could try to sidestep the IR problems by working on spacetimes with compact Cauchy slices (this isn't completely general, but it's a start). On the other hand, the historically first constructions of $\phi^4$ on Minkowski space by Glimm and Jaffe proceeded directly in real time without Euclidean methods (see e.g. Summers' review). Of course Minkowski space does have a lot of symmetry, so I don't really know what to make of this argument.
 A: Not a real answer, I'll be linking to non-rigorous discussion, but I hope it helps since it provides a discussion of the IR behavior of interacting theories.
The closest I know to your query is "Massless Thirring model in curved space: Thermal states and conformal anomaly", by Birrel and Davies back in 1978 (DOI: 10.1103/PhysRevD.18.4408). This paper discusses an interacting theory in 2D spacetime. However, it is a very particular theory, namely Massless Thirring Model. This model is a four-fermion interaction. Interestingly, the massless case is exactly solvable via Bethe Ansatz, and since 2D spacetime is conformally flat, you also have the exact solution for general 2D spacetime. With this in mind, the authors set up to show that for 2D Schwarzschild the interaction does not affect the thermal nature of the state. This was in the context of the recent discovery of Hawking radiation and trying to see whether interactions would alter the conclusions, which they show at least in this case it does not. For your purposes, it would indicate that the IR sector is tameable in 2D at least.
The other paper, more distant from what you seek, is "Interacting Quantum Fields Around a Black Hole", by, yes, Stephen Hawking in 1981 (DOI: 10.1007/BF01208279). In this paper, he studies $\lambda \phi^4$ theories only in Schwarzschild spacetime (4D, though given the actual spacetime I don't think this is an issue). The other issue is that because of the symmetries Hawking uses Euclidean methods. Now, this is far off from what you seek, but the reason I include it here is because of the discussion. You see, $\lambda \phi^4$ theories are known to exhibit phase transition in finite temperature flat spacetime. Hawking thus asks whether one could use the Hawking temperature to induce a phase transition in such a theory in the region close to the event horizon, where the temperature gets arbitrarily high. While he cannot provide a definite answer, his calculations suggest that in fact, you cannot. As of today, I know of no one who has provided a concrete solution here, but if Hawking is correct that would suggest that there are important differences between the flat thermal state and the Schwarzschild one, and an example of interesting behavior in the IR sector for interacting theories in curved spacetimes.
