Suppose that $\Sigma$ is a compact Riemann surface with boundary and that $f: \Sigma \rightarrow \mathbb{C}$ is holomorphic*. If $f$ is real-valued along $\partial \Sigma$, is it necessarily true that $f$ is constant?
*I want to be careful about what I mean by "holomorphic" at the boundary. If $A \subset \{z \in \mathbb{C}: \text{Im}(z) \geq 0\}$, then a function $A \rightarrow \mathbb{C}$ is holomorphic if it extends to a holomorphic function on some open neighborhood of $A$ in $\mathbb{C}$.
I'm quite sure the solution is to fill the details in this direction :
From your Riemann surface $\Sigma$ with boundary there is a complex conjugate one $\Sigma^*$ with the same topological space and with sheaf of holomorphic functions
$\qquad $ $h : \Sigma\to \Bbb{C}$ is holomorphic iff $\overline{h}:\Sigma^*\to \Bbb{C}$ is holomorphic.
The two can be glued as topological spaces along the boundary to give a space $X=\Sigma\cup \Sigma^*$. It has a sheaf of holomorphic functions which is $$\{ g_1\cup g_2, g_1\in Hol(\Sigma),g_2\in Hol(\Sigma^*),\ g_1|_{\partial \Sigma}=g_2|_{\partial \Sigma^*}\}$$
If your function $f$ is non-constant then the sheaf is non-trivial on the glued boundaries so this makes $X$ into a (true) compact Riemann surface (*), but then $f\cup \overline{f}$ is globally holomorphic $X\to \Bbb{C}$ thus constant by the maximum modulus principle.