  The double tensor multiplet of D=4, N=2 supersymmetry, relevant to type IIB
superstring vacua, is derived and its gauge invariant and N=2 supersymmetric
interactions are analysed, both self-interactions and interactions with vector
multiplets and hypermultiplets. Using deformation theory, it is shown that the
lowest dimensional nontrivial interaction vertices of this type have dimension
5 and all dimension 5 vertices are determined. They give rise to new N=2
supersymmetric gauge theories of the `exotic' type which are local but
nonpolynomial in some of the fields and coupling constants. Explicit examples
of such models are constructed.
