  The moduli space ${\rm M}_{d}$, of complex rational maps of degree $d \geq
2$, is a connected complex orbifold which carries a natural real structure,
coming from usual complex conjugation. Its real points are the classes of
rational maps admitting antiholomorphic automorphisms. The locus of the real
points ${\rm M}_{d}({\mathbb R})$ decomposes as a disjoint union of the loci
${\rm M}_{d}^{\mathbb R}$, consisting of the real rational maps, and ${\mathcal
P}_{d}$, consisting of the pseudo-real ones. We obtain that, both ${\rm
M}_{d}^{\mathbb R}$ and ${\rm M}_{d}({\mathbb R})$, are connected and that
${\mathcal P}_{d}$ is disconnected. We also observe that the group of
holomorphic automorphisms of a pseudo-real rational map is either trivial or a
cyclic group. For every $n \geq 1$, we construct pseudo-real rational maps
whose group of holomorphic automorphisms is cyclic of order $n$. As the field
of moduli of a pseudo-real rational map is contained in ${\mathbb R}$, these
maps provide examples of rational maps which are not definable over their field
of moduli. It seems that these are the only explicit examples in the literature
(Silverman) of rational maps which cannot be defined over their field of
moduli. We provide explicit examples of real rational maps which cannot be
defined over their field of moduli. Finally, we also observe that every real
rational map, which admits a model over the algebraic numbers, can be defined
over the real algebraic numbers.
