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Let me give a version of the question in the comment:

Let $X$ be a curve of genus $g$ with a real separting involution, and conisder the map $Sym^n(X_+)\to Jac^n(X)$.

For wich $n$ is this map surjective? Or, in other words, what is the minimal number of poles of a meromorphic function with poles in $X_+$ that garanties that zeros can happen at any collection of points?

This sounlds like a very nice question. In the case $g=1$ you can always take $n=2$. Also for any $g$ you sould take $n>g$ because $Sym^g(X)$ maps to $Jac^g(X)$ with degree $1$.

Added. The notation $Sym^n(X)$ means the symmetric power of $X$. Let me explain also why what is above is a reformulation of the original question. Indeed, a divisor $\sum_i x_i-\sum_i y_i$ on $X$ is a divisor of a meromorfic function iff it represent zero in $Jac^0(X)$. So if we want to chose arbitraly zeros $x_i$ of a meromorphic function $f$ keeping the poles $y_i$ in $X_+$ it is enouth to know that $\sum_i y_i$ can take any value in $Jac^n(X)$ (to cancel the point $\sum_i x_i$). This is eactly the condition that $Sym^n(X_+)\to Jac^n(X)$ is surjective.

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Liran, your

Let me give a version of the question seems to in the comment:

Let $X$ be a bit vague..curve of genus $g$ with a real separting involution, and conisder the map $Sym^n(X_+)\to Jac^n(X)$. Could you add For wich $n$ is this map surjective? Or, in other words, what is the minimal number of poles of a bit more detailsmeromorphic function with poles in $X_+$ that garanties that zeros can happen at any collection of points?

This sounlds like a very nice question. In the case $g=1$ you can always take $n=2$. Also for any $g$ you sould take $n>g$ because $Sym^g(X)$ maps to $Jac^g(X)$ with degree $1$.

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