  This paper is a companion to our earlier work arXiv:0710.3440 in which the
projective superspace formulation for matter-coupled simple supergravity in
five dimensions was presented. For the minimal multiplet of 5D N=1 supergravity
introduced by Howe in 1981, we give a complete solution of the Bianchi
identities. The geometry of curved superspace is shown to allow the existence
of a large family of off-shell supermultiplets that can be used to describe
supersymmetric matter, including vector multiplets and hypermultiplets. We
formulate a manifestly locally supersymmetric action principle. Its natural
property turns out to be the invariance under so-called projective
transformations of the auxiliary isotwistor variables. We then demonstrate that
the projective invariance allows one to uniquely restore the action functional
in a Wess-Zumino gauge. The latter action is well-suited for reducing the
supergravity-matter systems to components.
