  In this article, we prove that a smooth projective complex surface $X$ which
is regular (i.e. such that $h^1(X,\mathcal O_X)=0$) and which has a
$\mathbb{R}$-divisor $\Delta$ such that $(X,\Delta)$ is a KLT Calabi-Yau pair
has finitely many real forms up to isomorphism. For this purpose, we construct
a complete CAT(0) metric space on which $\text{Aut }X$ acts properly
discontinuously and cocompactly by isometries, using Totaro's Cone Theorem.
Then we give an example of a smooth rational surface with finitely many real
forms but having a so large automorphism group that our previous result (see
https://arxiv.org/abs/1409.3490) does not predict this finiteness.
