  We introduce real structures on $L$-twisted Higgs pairs over a compact
Riemann surface equipped with an anti-holomorphic involution, and prove a
Hitchin--Kobayashi correspondence for them. Real $G$-Higgs bundles, where $G$
is a real form of a connected semisimple complex affine algebraic group
$G^{\mathbb{C}}$, constitute a particular class of examples of these pairs. The
real structure in this case involves a conjugation of $G^{\mathbb{C}}$
commuting with the one defining the real form $G$. We establish a homeomorphism
between the moduli space of real $G$-Higgs bundles and the moduli space of
compatible representations of the orbifold fundamental group of $X$. Finally,
we show how real $G$-Higgs bundles appear naturally as fixed points of certain
anti-holomorphic involutions of the moduli space of $G$-Higgs bundles, that are
constructed using the real structures on $G^{\mathbb{C}}$ and $X$.
