  We study actions of discrete subgroups $\Gamma$ of semi-simple Lie groups $G$
on associated oriented flag manifolds. These are quotients $G/P$, where the
subgroup $P$ lies between a parabolic subgroup and its identity component. For
Anosov subgroups $\Gamma\subset G$, we identify domains in oriented flag
manifolds by removing a set obtained from the limit set of $\Gamma$, and give a
combinatorial description of proper discontinuity and cocompactness of these
domains. This generalizes analogous results of Kapovich-Leeb-Porti to the
oriented setting. We give first examples of cocompact domains of discontinuity
which are not lifts of domains in unoriented flag manifolds. These include in
particular domains in oriented Grassmannians for Hitchin representations, which
we also show to be nonempty. As a further application of the oriented setup, we
give a new lower bound on the number of connected components of $B$-Anosov
representations of a closed surface group into $\mathrm{SL}(n,\mathbb{R})$.
