Separating semigroup of genus 4 curves

A rational function on a real algebraic curve $C$ is called separating if it takes real values only at real points. Such a function defines a covering $\mathbb R C\to\mathbb{RP}^1$. Let $c_1,\dots,c_r$ be the connected components of $\mathbb R C$. M. Kummer and K. Shaw defined the separating semigroup of $C$ as the set of all sequences $(d_1(f),\dots,d_r(f))$ where $f$ is a separating function, and $d_i(f)$ is the degree of the restriction of $f$ to $c_i$.
In the present paper, we describe the separating semigroups of all genus 4 curves. For the proofs, we consider the canonical embedding of $C$ into a quadric $X$ in $\mathbb P^3$, and apply Abel's theorem to 1-forms on $C$ obtained as Poincaré residues of certain meromorphic 2-forms.