  Starting from the 48+48 component multiplet of supercurrents for a rigid N=2
tensor multiplet in four spacetime dimensions, we obtain the transformation of
the linearized supergravity multiplet which couples to this supercurrent
multiplet. At the linearized level, this 48+48 component supergravity multiplet
decouples into the 24+24 component linearized standard Weyl multiplet and a
24+24 component irreducible matter multiplet containing a real scalar field. By
a consistent application of the supersymmetry algebra with field dependent
structure constants appropriate to N=2 conformal supergravity, we find the full
transformation law for this multiplet in a conformal supergravity background.
By performing a field redefinition and switching off the conformal supergravity
background, the multiplet is equivalent to the one introduced by Howe et al in
flat space as a constrained real scalar superfield. We present a set of
constraints which can be consistently imposed on this multiplet to obtain a
restricted minimal 8+8 off-shell matter multiplet. We also show as an example
the precise embedding of the tensor multiplet inside this multiplet.
