  We investigate representations of Coxeter groups into
$\mathrm{GL}(n,\mathbb{R})$ as geometric reflection groups which are convex
cocompact in the projective space $\mathbb{P}(\mathbb{R}^n)$. We characterize
which Coxeter groups admit such representations, and we fully describe the
corresponding spaces of convex cocompact representations as reflection groups,
in terms of the associated Cartan matrices. The Coxeter groups that appear
include all infinite, word hyperbolic, irreducible Coxeter groups; for such
groups the representations as reflection groups that we describe are exactly
the projective Anosov ones. We also obtain a large class of nonhyperbolic
irreducible Coxeter groups, thus providing many examples for the theory of
nonhyperbolic convex cocompact subgroups in $\mathbb{P}(\mathbb{R}^n)$
developed in arXiv:1704.08711.
