Suppose we are given a $2n$-dimensional real vector space $V$, along with a complex structure $J:V\to V$. For $k\le n$, a $k$-subspace $W\subset V$ is called $J$-totally real if $W\cap JW = 0$. Consider $TR_k$ to be the set of all totally real $k$-subspaces of $V$. We can think of $TR_k$ as totally real Grassmannian.
Is the space $TR_k$ well understood in the literature? For example, is this a homogeneous space like the other well-known Grassmannians?
Any comment is appreciated. Thanks!