In the real analytic category, are the fibers of a proper submersion isomorphic?

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.