A fig grower in Hollywood, Florida has posted "Sweet Alabama Fig" trees on Craigslist.  A quick search yield no results for this variety.  Is this a Celeste or Brown Turkey?  Or is it a variety by that name?  Here is the Craigslist post http://miami.craigslist.org/brw/grd/3796894721.html

Since I had no replies here, I called the Craigslist poster who sells the "Sweet Alabama Fig".  The poster said the tree comes from his father's property in Alabama.  He said he didn't know the variety, but the "Mystery X" fig sold by TyTy Nursery is close.  Since that nursery has a horrible online reputation, I hesitated to buy one.  But he has no connection to TyTy and is only guessing it is a Mystery X, I'll pick up a couple of his trees. 

If they call it a sweet fig it is probably a Celeste which is also know as Sugar fig.  They look like some pretty nice trees.