Orbit duality in ind-varieties of maximal generalized flags

We extend Matsuki duality to arbitrary ind-varieties of maximal generalized flags, in other words, to any homogeneous ind-variety $\mathbf{G}/\mathbf{B}$ for a classical ind-group $\mathbf{G}$ and a splitting Borel ind-subgroup $\mathbf{B}\subset\mathbf{G}$. As a first step, we present an explicit combinatorial version of Matsuki duality in the finite-dimensional case, involving an explicit parametrization of $K$- and $G^0$-orbits on $G/B$. After proving Matsuki duality in the infinite-dimensional case, we give necessary and sufficient conditions on a Borel ind-subgroup $\mathbf{B}\subset\mathbf{G}$ for the existence of open and closed $\mathbf{K}$- and $\mathbf{G}^0$-orbits on $\mathbf{G}/\mathbf{B}$, where $\left(\mathbf{K},\mathbf{G}^0\right)$ is an aligned pair of a symmetric ind-subgroup $\mathbf{K}$ and a real form $\mathbf{G}^0$ of $\mathbf{G}$.