  In this paper we first formulate a dually gauged harmonic map model,
suggested from a product Abelian Higgs field theory arising in
impurity-inspired field theories, and obtain a new BPS system of equations
governing coexisting vortices and antivortices, which are topologically
characterized by the first Chern class of the underlying Hermitian bundle and
the Thom class of the associated dual bundle. We then establish existence and
uniqueness theorems for such vortices. For the equations over a compact
surface, we obtain necessary and sufficient conditions for the existence of
solutions. For the equations over the full plane, we obtain all finite-energy
solutions. Besides, we also present precise expressions giving the values of
various physical quantities of the solutions, including magnetic charges and
energies, in terms of the total numbers of vortices and antivortices, of two
species, and the coupling parameters involved.
