Likely a Brown Turkey or Celeste.  It came to me from my step mother and she grew up eating this fig on her family's propery in Sunflower, Mississippi.  The mother tree is quite old.  Until I eat the figs, and research opinions on it to better know what it is, that's the best I got for now.  It is a guess as to the two varieties I offered up as possibles because they were prevalent across the South until somewhat recently.