Morgan and Shalen "Free action of surface groups on R-trees" 1989 shows that surface groups (genus at least 2) act freely on some real trees (R-trees). Their proof seems to be non-constructive, requiring the Baire category theorem. But the object they are constructing sounds quite simple. Their Proposition 17 says:
On every closed non-exceptional hyperbolic surface, there exists a measured geodesic lamination whose leaves and complementary regions are all simply connected.
Does anyone have a picture of this lamination on the genus 2 surface? And, using that, a picture of the R-tree that the genus 2 surface group acts freely on?