  We obtain a characterization of Maximal and Galois-Maximal $C_2$-spaces
(including real algebraic varieties) in terms of
$\operatorname{RO}(C_2)$-graded cohomology with coefficients in the constant
Mackey functor $\underline{\mathbf{F}}_2$, using the structure theorem of
\cite{clover_may:structure_theorem}. Other known characterizations, for
instance in terms of equivariant Borel cohomology, are also rederived from
this. For the particular case of a smooth projective real variety $V$,
equivariant Poincar\'{e} duality from
\cite{pedro&paulo:quaternionic_algebraic_cycles} is used to deduce further
symmetry restrictions for the decomposition of the
$\operatorname{RO}(C_2)$-graded cohomology of the complex locus $V(\mathbf{C})$
given by the same structure theorem. We illustrate this result with some
computations, including the $\operatorname{RO}(C_2)$-graded cohomology with
$\underline{\mathbf{F}}_2$ coefficients of real $K3$ surfaces.
