  A real toric space is a topological space which admits a well-behaved
$\mathbb{Z}_2^k$-action. Real moment-angle complexes and real toric varieties
are typical examples of real toric spaces. A real toric space is determined by
a pair of a simplicial complex $K$ and a characteristic matrix $\Lambda$.
  In this paper, we provide an explicit $R$-cohomology ring formula of a real
toric space in terms of $K$ and $\Lambda$, where $R$ is a commutative ring with
unity in which $2$ is a unit. Interestingly, it has a natural $(\mathbb{Z}
\oplus \operatorname*{row} \Lambda)$-grading. As corollaries, we compute the
cohomology rings of (generalized) real Bott manifolds in terms of binary
matroids, and we also provide a criterion for real toric spaces to be
cohomology symplectic.
