  We show that it is possible to distinguish between different off-shell
completions of supergravity at the on-shell level. We focus on the comparison
of the ``new minimal'' formulation of off-shell four-dimensional N=1
supergravity with the ``old minimal'' formulation. We show that there are
3-manifolds which admit supersymmetric compactifications in the new-minimal
formulation but which do not admit supersymmetric compactifications in other
formulations. Moreover, on manifolds with boundary the new-minimal formulation
admits ``singleton modes'' which are absent in other formulations.
