Triality is a classical notion in geometry that arose in the context of the Lie groups of type D4. 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 L2 (q3) has maps that admit Wilson trialities but no dualities.