  We present a novel technique well suited to study the ground state of
inhomogeneous fermionic matter in a wide range of different systems. The system
is described using a Fermionic Shadow wavefunction (FSWF) and the energy is
computed by means of the Variational Monte Carlo technique. The general form of
FSWF is useful to describe many--body systems with the coexistence of different
phases as well in the presence of defects or impurities, but it requires
overcoming a significant sign problem. As an application, we studied the energy
to activate vacancies in solid $^3$He.
