While I was reading these notes, I've found the following version of the so called "Riemann existence theorem".
Do you know any reference for the proof of this statement?
Remark: In the algebraic setting, the result is very easy to show because the sheaf of meromorphic functions $\mathscr M_X$ is replaced by the locally constant sheaf given by the function field $K(X)$ (see this question). I think that the key word here is "GAGA"but I don't know exactly how to use it.
Using Riemann Roch, for example, you can find an embedding of $X$ in some $\Bbb P^N$. Similarly, for sufficiently large $k\in\Bbb N$, the line bundle (invertible sheaf) $\mathscr F\otimes \mathscr O(k)$ has a nontrivial global holomorphic section $s$. Letting $\sigma$ be a nontrivial section of $\mathscr O(k)$, i.e., a homogeneous polynomial of degree $k$ (restricted to $X$), $s/\sigma$ is a global meromorphic section of $\mathscr F$.