  The moduli space of smooth real binary octics has five connected components.
They parametrize the real binary octics whose defining equations have 0, 1,
..., 4 complex-conjugate pairs of roots respectively. We show that the
GIT-stable completion of each of these five components admits the structure of
an arithmetic real hyperbolic orbifold. The corresponding monodromy groups are,
up to commensurability, discrete hyperbolic reflection groups, and their
Vinberg diagrams are computed. We conclude with a simple proof that the moduli
space of GIT-stable real binary octics itself cannot be a real hyperbolic
orbifold.
