  We consider Dirichlet domains for Anosov subgroups $\Gamma$ of semisimple Lie
groups $G$ acting on the associated symmetric space $G/K$. More precisely, we
consider certain Finsler metrics on $G/K$ and a sufficient condition so that
every Dirichlet domain for $\Gamma$ is finite-sided in a strong sense. Under
the same condition, the group $\Gamma$ admits a domain of discontinuity in a
flag manifold where the Dirichlet domain extends to a compact fundamental
domain. As an application we show that Dirichlet-Selberg domains for $n$-Anosov
subgroups $\Gamma$ of $SL(2n,\mathbb{R})$ are finite-sided when all singular
values of elements of $\Gamma$ diverge exponentially in the word length. For
every $d\ge 3$, there are projective Anosov subgroups of $SL(d,\mathbb{R})$
which do not satisfy this property and have Dirichlet-Selberg domains with
infinitely many sides. More generally, we give a sufficient condition for a
subgroup of $SL(d,\mathbb{R})$ to admit a Dirichlet-Selberg domain whose
intersection with an invariant convex set is finite-sided.
