Might be Araujia sericifera

Whenever I hear "vine" and "milkweed" I can't help to think of Pale Swallow wort (Cynanchum rossicum). Vile, invasive plant. There is a black swallow wort species as well. Compare and see what you think.

FataMorgana

Here is a link that might be useful: Google images - Cynanchum pods

Those were both good possibilities. However, think I found the match. We have Gonolobus suberosus var. granulatus (Angular-fruit Milkvine) in this area. A photo of that species is shown below. Another variant (var. suberosus) has more rounded pods like the one shown in the original post. That one grows in the southeastern US (FL, GA, NC, SC, VA), so isn't expected here. But it does match per Internet photos of the pods. Var. suberosus should have yellow and purple petals, instead of the greenish-yellow ones seen on our plants, so I'll confirm that this summer. I guess that I should mention that some botanists lump both of those into one species (Matelea gonocarpus).

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