If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
There has been some attempts at categorizing infinite-dimensional quantum physics.
While finite-dimensional quantum physics is pretty much categorized (symmetric monoidal dagger categories, see the works of Coecke and Abramsky), the infinite version is still in its infancy.
If a fully quantum category will be found, and showed to have more than one model, then I think that the idea of reality as an infinite-dimensional configuration space inhabited by an atemporal wave-function should be abandoned.
There are already examples of quantum theories with multiple interpretations, most famously AdS/CFT. Do these affect your view of realism?