  Traditionally, alternating links are studied with alternating diagrams on
$S^2$ in $S^3$. In this paper, we consider links which are alternating on
higher genus surfaces $S_g$ in $S_g \times I$. We define what it means for such
a link to be right-angled generalized completely realizable (RGCR) and show
that this property is equivalent to the link having two totally geodesic
checkerboard surfaces, equivalent to each checkerboard surface consisting of
one type of polygon, and equivalent to a set of restrictions on the link's
alternating projection diagram. We then use these diagram restrictions to
classify RGCR links according to the polygons in their checkerboard surfaces,
provide a bound on the number of RGCR links for a given surface of genus $g$,
and find an RGCR knot. Along the way, we answer a question posed by
Champanerkar, Kofman, and Purcell about links with alternating projections on
the torus.
