  Triality is a classical notion in geometry that arose in the context of the
Lie groups of type $D_4$. Another notion of triality, Wilson triality, appears
in the context of reflexible maps. We build a bridge between these two notions,
showing how to construct an incidence geometry with a triality from a map that
admits a Wilson triality. We also extend a result by Jones and Poulton, showing
that for every prime power $q$, the group ${\rm L}_2(q^3)$ has maps that admit
Wilson trialities but no dualities.
