Thanks to FloriFloydian Jeff, these things have now been ID’d as “air potato.” They’re not from around here (originated in Africa and Asia) but have quite taken over Florida’s native forests in places. See this nature.org piece about air potato.
The “nuts”, as I thought, are vegetative “propagules” by which these things spread, along with underground tubers that persist after burning or cutting the above-ground parts. Once established, these things are the devil to get rid of.
If I’d seen the source, I would have recognized the genus: Dioscorea. It shows a close resemblance to our southern US “wild yam” in the same genus.
The edibility of these tuberous growths is apparently debatable. Wikipedia claims this is the most widely-consumed yam species world-wide. The wild ones, like those growing near our friends’ place in Sarasota, are considered poisonous because they contain a steroid similar to that used in hormonal contraceptives.
As US climate warms, expect this one eventually to survive farther and farther north. Kudzu, meet Air Potato.