Good timing, I just found what I think is the actual species name for this plant a few days ago when I was searching for something else. It appears as though this plant is still a Melastomataceae, not of the genus Bertolonia, rather it is Clidemia hirta. I was a little surprised to find that it is considered an extremely invasive species in tropical areas, considering it is somewhat difficult to propagate from unrooted cuttings. This page describes its natural history and invasive attributes fairly well. Didn't know it could develop vine-like growth, but it probably requires space and maturity for that.