Let $m>0$ be a natural number. Consider the following semisimple algebraic groups over ${\mathbb{R}}$: $$ G={\mathrm{SU}}(2m,4m),\ \ H={\mathrm{SU}}(2m,2m)\times{\mathrm{SU}}(2m). $$ We embed $H$ into $G$ "block-diagonally" and consider the homogeneous space $X=G/H$, which is an affine variety over ${\mathbb{R}}$. Using Galois cohomology, I can compute the number of connected components of $X({\mathbb{R}})$: $$ \#\pi_0(X({\mathbb{R}}))=m+1. $$ Probably it is possible to obtain this result from classical algebraic geometry.
Question. How can one obtain this result from classical algebraic geometry, without Galois cohomology?