In the real analytic category, are the fibers of a proper submersion isomorphic? 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?
 A: There exists an analytic Riemann metric on a real-analytic manifold, which follows from embeddability of real analytic manifolds (see  The Analytic Embedding of Abstract Real-Analytic Manifolds Charles B. Morrey, Jr.) - maybe can also be proved easier.
So, you can probably just take orthogonal connection.
A: I think Ehresmann's theorem does hold in the real-analytic category. A proof via tubular neighborhoods can be repeated in the real-analytic context since it makes no use of partitions of unity. Here's a sketch.
By Lev's answer, a real-analytic manifold admits a real-analytic embedding into Euclidean space with its standard real-analytic structure. I think this implies a real-analytic manifold admits a real-analytic Riemannian metric.
The exponential map of this metric is also real-analytic (as a solution to analytic ODE), so an analytically embedded submanifold admits a real-analytic tubular neighborhood. We may then repeat this proof of Ehresmann's theorem using tubular neighborhoods.
Remark. This comment along with several other places in the literature assert Ehresmann's theorem fails in the real-analytic category but provide no examples.
