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.
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