Ehresmann's theorem says that a proper smooth submersion is a fiber bundle. The proofs I know rely on the existence of connections locally on the base, and this is furnished by partitions of unity.

This question gives a counterexample in the holomorphic category which is probably classic (elliptic curves), but I don't know any of the story.

I am wondering about the real analytic category. Are the fibers of a proper real analytic submersion isomorphic? If not, will it be locally trivial (in the real analytic category) when they are isomorphic, as in the linked question?