  The moduli space of smooth real plane quartic curves consists of six
connected components. We prove that each of these components admits a real
hyperbolic structure. These connected components correspond to the six real
forms of a certain hyperbolic lattice over the Gaussian integers. We will study
this Gaussian lattice in detail. For the connected component that corresponds
to maximal real quartic curves we obtain a more explicit description. We
construct a Coxeter diagram that encodes the geometry of this component.
