Let $A$ be a principally polarised abelian variety over $\mathbb{C}$ of dimension $g$. Let $K$ be the field of moduli of $A$.

Proposition.$A$ has a model defined over an extension $L$ of $K$ such that $[L:K]$ is bounded above by a function of $g$.

This follows from the fact that "principally polarised abelian varieties of dimension $g$ with full level-$3$ structure" have a fine moduli space $\mathcal{A}_g(3)$. Hence the proposition holds with the bound being the degree of the forgetful map $\mathcal{A}_g(3) \to \mathcal{A}_g$.

Is the proposition written down anywhere that I could cite? I assume it is well-known but I have not been able to find it. A referee has asked me to provide a reference.