Let $M$ $N$ $P$ be complex manifolds, and let $$f: M\rightarrow N, g: N\rightarrow P$$ be $C^\infty$ maps with $g$ and $g\circ f$ holomorphic, and with $dg$ never degenerate. It's easy, then, to see that $f$ is holomorphic:
$f$ is holomorphic iff $df$ commutes with "multiplication-by-$i$", which I'll call $m$. We have
$$dg \circ df \circ m= m \circ dg \circ df = dg \circ m \circ df $$ Where the first inequality is due to $g\circ f$ holomorphic, and the second inequality to $g$ holomorphic. Since $dg$ is injective, we can cancel it in the equality to get that $df$ commutes with $m$.
This argument is given in Knapp's Lie Groups: Beyond an Introduction as a step toward proving the exponential map of a complex Lie group is holomorphic (using Ado's theorem to reduce to the case where the group is inside $GL(n, \mathbb{C})$). Toward proving that any $C^\infty$ Lie group admits a real analytic structure, Knapp cites the analogous result for real analytic manifolds, i.e. the result gotten from the statement above by replacing "complex manifold" with "real analytic manifold" and "holomorphic" with "real analytic". However all he says is that "the analogous result is valid." I'm unable to fill in the proof.
I should mention that Knapp's definition of a real analytic function $f: \mathbb{R}^m \rightarrow \mathbb{R}^n$ $ (U \subset \mathbb{R}^m )$ is that f extends to a holomorphic function on some open set of $\mathbb{C}^m$ around any point in U.
There is no difference between Knapp's definition and usual definition. Real analytic functions are all obtained in this way. In local sense, we can just consider real analytic maps between regions on $\mathbb{R}^n$.
Suppose $f : U \to V$ and $g: V \to W$ are real analytic. Naturally we just have to extend $g \circ f$ and $g$ to $H$ and $G$ as holomorphic functions from some domains in $\mathbb{C}^n$. Set $F = G^{-1} \circ H$ to be a holomorphic function. Here $dG$ to be non-degenerate since $dg$ is non-degenerate and $G$ is holomorhic.
Finally, $F$ restricts back to a real analytic function on $U$. This function is $f$.