Lately I came across the following lemma (in Michor "Topics in Differential Geometry", p. 200, Lemma 17.2):
Let $p: M\rightarrow N$ be a surjective submersion which is proper and let $N$ be connected. Then $(M, p, N)$ is a (smooth) fibre bundle.
The proof relies heavily on partition of unity, which does not exist in the (real-)analytic case. Therefore, my question is:
Are there sufficient conditions such that the lemma is true in the (real-)analytic category?
As requested, I will address the complex-analytic version of this question. In short, the classical theory of moduli spaces is build around such examples.
The key (trivial) observation is that if $\pi: E\to M$ is a (locally holomorphically trivial) holomorphic bundle with connected base manifold $M$, then for all $x\in M$ the fibers $\pi^{-1}(X)$ are all biholomorphic to each other. (To see this, first observe that this holds locally on $M$ and then it holds globally on $M$ since $M$ is connected).
Now, let $M_g$ denote the moduli space of compact Riemann surfaces of genus $g\ge 1$. The complex space $M_g$ is an orbifold (a Deligne-Mumford stack) but at generic points $M_g$ is a complex manifold of dimension $3g-3$ (if $g\ge 2$) or $1$ if $g=1$. Genericity of $X\in M_g$ is understood as triviality of the group of automorphisms of $X$ if $g\ge 3$ and the equality of $Aut(X)$ to the order 2 group generated by the hyperelliptic involution if $g\le 2$.
Over $M_g$ we have the universal curve $\pi: U\to M_g$ whose fiber over a Riemann surface $X\in M_g$ is biholomorphic to $X$. The mapping $\pi$ is a holomorphic submersion over generic points of $M_g$. However, fibers $\pi^{-1}(X)$ are all pairwise non-isomorphic.
One can get more concrete examples by looking at the genus $g=1$ case, see e.g. these noted by Dick Hain. Here is another example (due to Kodaira). Let $X$ be a compact Riemann surface of genus $g\ge 2$, let $\alpha: X\to X$ be an automorphism without fixed points; consider the diagonal embedding $\Delta$ of $X$ into $X^2$ and let $A\subset X^2$ denote the graph of $\alpha$. Then $\Delta$ and $A$ are disjoint. There exists a 2-fold branched cover $p: M\to X^2$ ramified over $\Delta \cup A$. The composition $\pi$ of $p$ and the projection $X^2\to X$ is a holomorphic submersion. However, $\pi: M\to X$ is a locally nontrivial fibration. The fiber $\pi^{-1}(x)$ is a 2-fold branched cover over $X$ ramified over the points $x, \alpha(x)$.
Edit. As for local triviality in the real-analytic setting, it might be true. Grauert proved uniqueness of real-analytic structures on manifolds (with fixed underlying smooth structure), see
H. Grauert, H., 1958, On Levi'sproblemand the imbedding o freal analytic manifolds, Annals Math. 68, 460-472.
More precisely, Grauert proved that every real-analytic manifold embeds in some $R^N$, while Whitney proved earlier that in the embedded setting the real-analytic structure is unique (using Weierstrass approximation theorem).
Maybe Grauert-Whitney proof can be used to prove local triviality for real-analytic proper submersions.
Edit. 1. The reference to Whitney's proof is:
H. Whitney, Differentiable manifolds. Ann. of Math. (2) 37 (1936), no. 3, 645–680.
K. Shiga, Some aspects of real-analytic manifolds and differentiable manifolds. J. Math. Soc. Japan 16 (1964) 128–142.
Among other things, he reproves Whitney's theorem and has some results about proper analytic submersions which, I think, yield a positive answer to your original question. Namely, look at his Theorem 3 and apply it in your setting. We already know that $p: M\to N$ is locally trivial as $C^\infty$ bundle. In other words, there is a family of open subsets $U_i\subset N$ and $C^\infty$-diffeomorphisms $M_i:=p^{-1}(U_i)\to U_i\times F$, where $F$ is some fixed (real analytic) manifold. Shiga's theorem 3 gives you a real-analytic diffeomorphism $\phi_i: M_i\to U_i\times F$ which preserves the bundle structure. Thus, the maps $\phi_i$ provide a real-analytic local trivialization for $p$.