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)?