I'm wondering when a compact hyperbolic $n$-manifold ($n \geq 3$) can embed in a complex hyperbolic $n$-manifold as a real algebraic subvariety so that it is a component of the fixed point set of complex conjugation?

I suspect it might to be possible to answer this for arithmetic hyperbolic manifolds by an arithmetic construction. But I'm particularly interested in the 3-dimensional case, where most manifolds are not arithmetic.

One could vaguely hope to approach this using algebraic geometry. Make the manifold into a real algebraic variety, and then embed into a non-singular complex projective variety by resolving singularities. If this variety satisfies the Yau-Miyaoka inequality, then it is complex hyperbolic (see Theorem 1.3 of this paper for the statement and references). Obviously I have no idea how to achieve this though.

The restriction $n\geq 3$ is necessary, since in dimension 2 there are moduli spaces of hyperbolic structures, but compact complex hyperbolic surfaces are rigid and hence countable (the $1$-dimensional case is trivial).