I think Douglas had it right Ligodium Japonicum and it is invasive in FL. I think it is beautiful and wish I had one! And by the way, this is a great pic!
I'm afraid that is Japanese climbing fern. There is a native climbing fern that is not invasive, Lygodium palmatum, but the leaves look different. I know this is not what you want to hear, but I think you should remove this plant. It is on its way to becoming the next kudzu. It's a category 1 noxious weed in Florida and Georgia, a class B noxious weed in Alabama and a "severe threat" in South Carolina. Check out the photos of the destruction it can cause here:
Well it hasn't been invasive for the 40 years I have been here. Interesting how the change of 3 or 4 states can change a plant totally..
@Susan White It's already been reported as having escaped cultivation in North Carolina:
I really am intrigued by the texture, since I have mostly full sun, doubt that's one I'll need to worry about, but it sure is pretty.
It looks like one of the species of fern. However I have no idea of the name
It is lovely. I moved to my present house 8 years ago and discovered this beautiful, lacy vine. Hated to mess with it because it was so pretty. Now? Aaarrgh! I can't get rid of it!!! Constant battle!
@Susan White Ligodium Japonicum is invasive in wet areas but can be controlled in areas that have dry periods so if you are in a drier more mountainous area you should be fine