  The off-shell vector-tensor multiplet is considered in an arbitrary
background of N=2 vector supermultiplets. We establish the existence of two
inequivalent versions, characterized by different Chern-Simons couplings. In
one version the vector field of the vector-tensor multiplet is contained
quadratically in the Chern-Simons term, which implies nonlinear terms in the
supersymmetry transformations and equations of motion. In the second version,
which requires a background of at least two abelian vector supermultiplets, the
supersymmetry transformations remain at most linear in the vector-tensor
components. This version is of the type known to arise from reduction of tensor
supermultiplets in six dimensions. Our work applies to any number of
vector-tensor multiplets.
