  We have used an extended Scaled-Particle Theory that incorporates four-body
correlations through the fourth-order virial coefficient to analyse the
orientational properties of a fluid of hard right-angle triangles. This fluid
has been analysed by computer simulation studies, with clear indications of
strong octatic correlations present in the liquid-crystal phase, although the
more symmetric order tetratic phase would seem to be the most plausible
candidate. Standard theories based on the second virial coefficient are unable
to reproduce this behaviour. Our extended theory predicts that octatic
correlations, associated to a symmetry under global rotations of the oriented
fluid by $45^{\circ}$, are highly enhanced, but not enough to give rise to a
thermodynamically stable phase with strict octatic symmetry. We discuss
different scenarios to improve the theoretical understanding of the elusive
octatic phase in this intriguing fluid.
