Fundamental Polygon of Real Projective Plane
Solution 1:
If you're ok with the fundamental polygon being a polygon on a surface which isn't $\mathbb{R}^2$, then you can get a polygon with two edges on the sphere $S^2$.
The polygon is given by taking the northern hemisphere, and the edges are given by each a half of the equator, with the identification being the antipodal map on the circle defining the equator.
If you know how covering spaces work, this region is a fundamental region of the $2$-fold covering of the sphere on to the real projective plane.