The real quotient of the Cartwright-Steger surface

Borisov and Yeung showed that the Cartwright-Steger surface $X$ is defined over $\mathbb{Q}$, hence complex conjugation gives an antiholomorphic involution on $X$. We show that the orbit space $Y$ is a closed oriented smooth four-manifold homeomorphic to ${\mathbb{C}\mathbb{P}}^2\mathrm{\#}(S^1\times S^3)$, presenting $X$ as a double branched cover $X\to Y$, with branch locus diffeomorphic to the connected sum of three copies of the real projective plane.