TITLE: Example of affine locally symmetric space
QUESTION [6 upvotes]: Who can give an example of an affine locally symmetric space that is not a Riemanian locally symmetric space?

REPLY [5 votes]: Having had hard time visualizing the example from the answer by Robert Bryant, I decided to make some graphs for it. They turned out so beautiful that I decided to share them.
Geodesics passing through $(0,1)$:

Geodesics passing through $(1,-0.3)$:

And for the "dual" example, geodesics passing through $(0,-0.1)$:

Geodesics passing through $(1,1)$: