  In the framework of the superconformal tensor calculus for 4D N=2
supergravity, locally supersymmetric actions are often constructed using the
linear multiplet. We provide a superform formulation for the linear multiplet
and derive the corresponding action functional using the ectoplasm method (also
known as the superform approach to the construction of supersymmetric
invariants). We propose a new locally supersymmetric action which makes use of
a deformed linear multiplet. The novel feature of this multiplet is that it
corresponds to the case of a gauged central charge using a one-form potential
not annihilated by the central charge (unlike the standard N=2 vector
multiplet). Such a gauge one-form can be chosen to describe a variant nonlinear
vector-tensor multiplet. As a byproduct of our construction, we also find a
variant realization of the tensor multiplet in supergravity where one of the
auxiliaries is replaced by the field strength of a gauge three-form.
