Is there a useful classification of the homogeneous spaces for real Lie groups?

Question

Let $G$ be a compact semisimple real Lie group. For the complex case there is a very deep theory connecting $G$-homogeneous spaces with irreducible representations of $G$. My question is:

1. Is there an analogous correspondence for the real case?

In case this is too general - I am interested in homogeneous spaces of either $G = \mathrm{U} (n)$, $\mathrm{SU} (n)$ or $\mathrm{PU} (n)$ (the unitary, special unitary and projective unitary groups, respectively). Since these are (real) linear algebraic groups, I add here a "weaker" question:

2. Is there any useful classification of the homogeneous spaces of real linear algebraic groups, or at least of the unitary groups in particular?