  In the prequel to this paper, two versions of Le Potier's strange duality
conjecture for sheaves over abelian surfaces were studied. A third version is
considered here. In the current setup, the isomorphism involves moduli spaces
of sheaves with fixed determinant and fixed determinant of the Fourier-Mukai
transform on one side, and moduli spaces where both determinants vary, on the
other side. We first establish the isomorphism in rank one using the
representation theory of Heisenberg groups. For product abelian surfaces, the
isomorphism is then shown to hold for sheaves with fiber degree 1 via
Fourier-Mukai techniques. By degeneration to product geometries, the duality is
obtained generically for a large number of numerical types. Finally, it is
shown in great generality that the Verlinde sheaves encoding the variation of
the spaces of theta functions are locally free over moduli.
