  The semi-inclusive deep inelastic electron scattering off transversely
polarized $^3$He, i.e. the process, $e + \vec{^3 {\rm He}} \to e' + h+X$, with
$h$ a detected fast hadron, is studied beyond the plane wave impulse
approximation. To this end, a distorted spin-dependent spectral function of a
nucleon inside an A=3 nucleus is actually evaluated through a generalized
eikonal approximation, in order to take into account the final state
interactions between the hadronizing system and the (A-1) nucleon spectator
one. Our realistic description of both nuclear target and final state is a
substantial step forward for achieving a reliable extraction of the Sivers and
Collins single spin asymmetries of the free neutron. To illustrate how and to
what extent the model dependence due to the treatment of the nuclear effects is
under control, we apply our approach to the extraction procedure of the neutron
single spin asymmetries from those measured for $^3$He for values of the
kinematical variables relevant both for forthcoming experiments at Jefferson
Lab and, with an exploratory purpose, for the future Electron Ion Collider.
