Let $\mathcal E$ be a complex elliptic curve and $S$ a non-empty finite subset of $\mathcal E$. We show the equality of two algebras of multivalued functions on $\mathcal E\smallsetminus S$: on the one hand, an algebra constructed using the functions $\tilde\Gamma$ introduced in arXiv:1712.07089 out of string theory motivations; on the other hand, the minimal algebra of holomorphic multivalued functions on $\mathcal E\smallsetminus S$ which is stable under integration, studied in arXiv:2212.03119; both algebras coincide with the algebra of multivalued holomorphic functions with unipotent monodromy on $\mathcal E\smallsetminus S$ and moderate growth at the points of $S$.