Relative periods of abelian differentials in a prescribed stratum

We characterise the elements of $H^1(S,Z,\mathbb{C})$, where $S$ is a closed surface and $Z\subset S$ is a finite set, that arise as the relative periods of an abelian differential in a given connected component of a stratum of their moduli space. This generalises a theorem obtained independently by Bainbridge, Johnson, Judge and Park and the author, and answers a question of Filip.