The Tarski-Seidenberg Theorem states that the polynomial image of a semi-algebraic set is semi-algebraic. A semi-algebraic subset of a Euclidean space $\Bbb{R}^n$ is by definition a finite union of subsets of the form $$ \{P_1=\dots=P_k=0, Q_1>0,\dots,Q_l>0\} $$ where $P_i$'s and $Q_j$'s belong to $\Bbb{R}[x_1,\dots,x_n]$. I wonder is there a general coordinate-free version of this theorem for morphisms of real varieties? (By a real variety I mean the set of $\Bbb{R}$-points of a variety over $\Bbb{R}$.)