Assuming that we live in a 10 dimensional spacetime, it is often informally argued that according to string theory we live in a product of Minkowski space and a 6-manifold, which because of the required compatibility with general relativity and with supersymmetry actually has to be a Calabi-Yau threefold (I hope what I say is more or less correct). Slightly more generally, it could be a fiber bundle over Minskowski space whose fiber is a Calabi-Yau threefold.

However, from these informal arguments it is not clear why or if we need to have the same Calabi-Yau variety at every point. Could it more generally be a smoothly varying family of (generically) Calabi-Yau manifolds over Minkowski space, or is there a physical reason why it most be locally trivial?