Curvature isn't necessary here; all we need is time-orientability, so we can even be more general than a Lorentzian manifold. We can achieve time-orientability by comparing how strictly we must constrain the degrees of freedom of, for example, an adiabatically expanding or contracting cloud of gas of non-interacting test particles below some critical mass-density such that expansion will carry on forever, rather than there being some eventual recontraction. This is perfectly doable in flat spacetime. It's essentially just a problem in statistical mechanics, as we can arrange time-orientability this way without having anything to do with relativity.

We don't really need time-orientability in relativity; it is perfectly reasonable to have solutions to the field equations which are static or stable periodic (and thus there is no clear past/future). Conversely, more generally we can get time-orientability in a wide variety of dimensions other than 3+1.

Relativity just tells us that where there is some global time-orientable feature, every observer will agree what's the past and what's the future of that feature. However, complex observers may have some internal degrees of freedom providing a local notion of time-orientability which could be unaligned with the global feature (and other observers' local features).

I don't see how any of this can relate to "consciousness" though. Also, our universe really doesn't admit backwards time travellers as far as we can tell, so whether and how the wider universe "corrects" observers who have different past/future orientations is really really really academic from a physics perspective. Sean Carroll's blog had a lot about that a decade or more ago, which you can probably dig out of preposterousuniverse.com or wherever.