  A Riemann surface $\mathcal{S}$ having field of moduli $\mathbb{R}$, but not
a field of definition, is called \emph{pseudoreal}. This means that
$\mathcal{S}$ has anticonformal automorphisms, but non of them is an
involution. We call a Riemann surface $\mathcal{S}$ \emph{plane} if it can be
described by a smooth plane model of some degree $d\geq4$ in
$\mathbb{P}^2_{\mathbb{C}}$.
  We characterize pseudoreal-plane Riemann surfaces $\mathcal{S}$, whose
conformal automorphism group $\operatorname{Aut}_+(\mathcal{S})$ is
$\operatorname{PGL}_3(\mathbb{C})$-conjugate to a finite non-trivial group $G$
that leaves invariant infinitely many points of $\mathbb{P}^2_{\mathbb{C}}$. In
particular, we show that such pseudoreal-plane Riemann surfaces exist only if
$\operatorname{Aut}_+(\mathcal{S})$ is cyclic of even order $n$ dividing the
degree $d$. Explicit examples are given, for any degree $d=2pm$ with $m>1$ odd,
$p$ is prime and $n=d/p$.
