  Let $X$ be a complex smooth algebraic variety admitting a symmetry $L$, that
is, an antiholomorphic automorphism of order two. If both, $X$ and $L$ are
defined over $\overline{\mathbb Q}$, then Koeck, Lau and Singerman showed the
existence of a complex smooth algebraic variety $Z$ admitting a symmetry $T$,
both defined over ${\mathbb R} \cap \overline{\mathbb Q}$, and of an
isomorphism $R:X \to Z$ so that $R \circ L \circ R^{-1}=T$. The provided proof
is existential and, if explicit equations for $X$ and $L$ are given over
$\overline{\mathbb Q}$, then it is not described how to get the explicit
equations for $Z$ and $T$ over ${\mathbb R} \cap \overline{\mathbb Q}$. In this
paper we provide an explicit rational map $R$ defined over ${\mathbb Q}$ so
that $Z=R(X)$ is defined over ${\mathbb R} \cap \overline{\mathbb Q}$ and with
$T=R \circ L \circ R^{-1}$ being the usual conjugation map.
