  We provide an experimental study of the existence of Herman rings in a
parametrized family of rational maps preserving antipodal points, and a
discussion of their properties on $\mathbb{RP}^2$. We study analytic maps of
the sphere that project to dianalytic maps on the nonorientable surface,
$\mathbb{RP}^2$. They have a known form and we focus on a subset of degree $3$
dianalytic maps and explore their dynamical properties. In particular, we focus
on maps for which the Fatou set has a Herman ring. We appeal to dynamical
properties of particular maps to justify our assertion that these Fatou
components are Herman rings and analyze the parameter space for this family of
maps.
