Semialgebraicity of the convergence domain of an algebraic power series

Given a power series in finitely many variables that is algebraic over the corresponding polynomial ring over a subfield of the reals, we show that its convergence domain is semialgebraic over the real closure of the subfield. This gives in particular that the convergence radius of a univariate Puiseux series that is algebraic in the above sense belongs to the real closure or is infinity.