What to call it (or any persistent caprifig outside of wasp territory) is a little more difficult. Smyrna caprifigs (short styled female flowers) are reported to be edible if there is no wasp and they are sprayed with growth hormones to stimulate ripening, it is the wasp eggs that cause the flowers to be dry. A persistent caprifig therefore will be edible if there is no wasp. To actually be an "edible persistent caprifig" it must have long styled female flowers, which the wasp cannot lay eggs in, so it will be edible even when visited by the wasp.
There is still lots to learn about this, edible persistent caprifigs genetics are not explained by any research I have seen... the conventional wisdom is that the gene for stamens is tightly linked to the gene for short styles female flowers.