  We show that each connected component of the moduli space of smooth real
binary quintics is isomorphic to an arithmetic quotient of an open subset of
the real hyperbolic plane. Moreover, our main result says that the induced
metric on this moduli space extends to a complete real hyperbolic orbifold
structure on the space of stable real binary quintics. This turns the moduli
space of stable real binary quintics into a real two-dimensional ball quotient,
whose fundamental domain is given by the hyperbolic triangle of angles $\pi/3,
\pi/5$ and $\pi/10$, and whose fundamental group is non-arithmetic.
