EDIT: Better rewording by Chris White:
Is it possible to have a theory that treats both GR and QFT (e.g. QFT on a curved spacetime dynamically influenced by the standard QFT fields)? Is such a theory at least self-consistent (even if it does not apply to nature)? Or is there some fundamental incompatibility we run into without even trying to quantize GR (or perhaps we are somehow forced to quantize GR for consistency)?
Best Answer
This formalism exists, and is called semiclassical gravity. You can calculate effects using it, most notably, the existence of Hawking radiation. As an exact solution, it is somewhat unsatisfactory for these reasons:
Anyway, if you're interested in the topic, Wald wrote an (extremely technical) book on it: http://press.uchicago.edu/ucp/books/book/chicago/Q/bo3684008.html