Real algebraic GM$\mathbb Z$-surfaces

We prove necessary and sufficient conditions for a real algebraic surface to be a $\operatorname{GM}\mathbb Z$-surface. We calculate the Neron–Severi group $\operatorname{NS}(X)$, the Brauer group $\operatorname{Br}(X)$ and the algebraic cohomology group $H_a^1(X(\mathbb R),\mathbb F_2)$, where $X$ is a real projective surface. We also prove Nikulin's congruence for an arbitrary orientable $M$-surface