  The moduli space ${\mathcal{M}}_{g}$, of genus $g\geq2$ closed Riemann
surfaces, is a complex orbifold of dimension $3(g-1)$ which carries a natural
real structure i.e. it admits an anti-holomorphic involution $\sigma$. The
involution $\sigma$ maps each point corresponding to a Riemann surface $S$ to
its complex conjugate $\overline{S}$. The fixed point set of $\sigma$ consists
of the isomorphism classes of closed Riemann surfaces admitting an
anticonformal automorphism. Inside $\mathrm{Fix}(\sigma)$ is the locus
${\mathcal{M}}_{g}(\mathbb{R})$, the set of real Riemann surfaces, which is
known to be connected by results due to P. Buser, M. Sepp\"{a}l\"{a} and R.
Silhol. The complement $\mathrm{Fix}(\sigma)-{\mathcal{M}}_{g}(\mathbb{R})$
consists of the so called pseudo-real Riemann surfaces, which is known to be
non-connected. In this short note we provide a simple argument to observe that
$\mathrm{Fix}(\sigma)$ is connected.
