Let $\mathbf{G}$ be a semisimple $\mathbb{Q}$-algebraic group, $\mathbf{G}(\mathbb{R})^+$ the connected component (for the Euclidean topology) containing the identity, $\Gamma\subset \mathbf{G}(\mathbb{R})^+\cap\mathbf{G}(\mathbb{Q})$ an arithmetic lattice, $K\subset \mathbf{G}(\mathbb{R})^+$ a maximal compact subgroup. The quotient $X:=\Gamma\backslash (\mathbf{G}(\mathbb{R})^+/K)$ is a so-called "locally symmetric space". It is a real-analytic space. It is known, by Baily-Borel, that if $\mathbf{G}_\mathbb{R}$ is of Hermitian type then $X$ has the structure of a quasi-projective complex algebraic variety, functorial in the data.
If $\mathbf{G}_\mathbb{R}$ is not of Hermitian type, is there a "natural" way to endow $X$ with a real algebraic structure?
