Can there exist smooth, proper $X_1,X_2/\mathbb Z_p$ such that their generic fibers are isomorphic but their reductions mod $p$ are not? Are there examples if we insist that the special fibers are distinct even over $\overline{\mathbb F}_p$?
An obstruction is that the two reductions should have the same etale cohomology (by proper base change) and tame geometric etale fundamental group (by Grothendieck's comparison theorem).
