Let $X,Y$ be two normal algebraic surfaces (for instance projective) and let $\varphi\colon X\dashrightarrow Y$ be a birational map which restricts to an isomorphism $(X\setminus F)\to (Y\setminus G)$ where $F\subset X,G\subset Y$ are finite subsets. Does it follow that $\varphi$ is an isomorphism?

(This is true at least when $X$ and $Y$ are smooth).