Let $X$ be an algebraic variety over $K$ and let $\varphi:W\rightarrow{T^{\rm an}}$ be an analytic moment map defined on an open subset $W$ of ${X^{\rm an}}$. For every $x\in W$, there is a very affine open subset $U$ of $X$ with an algebraic moment map $\varphi^{\prime}:U\rightarrow T$ and an open neighbourhood $V$ of $x$ in ${U^{\rm an}}\cap W$ such that ${\varphi_{\rm trop}}=\varphi^{\prime}_{\rm trop}$ on $V$.