  The final state interaction in DIS of leptons off a nucleus A, due to the
propagation of the struck nucleon debris and its hadronization in the nuclear
environment is considered. The effective cross section of such a partonic
system with the nucleons of the medium and its time dependence are estimated on
the basis of a model which takes into account the production of hadrons due to
the breaking of the color string and to gluon radiation. It is shown that the
details of the propagation of the partonic system in the nuclear environment
can be thoroughly investigated by the semi-inclusive process A(e,e'(A-1))X, in
which the scattered lepton is detected in coincidence with the nucleus (A-1) in
low energy and momentum states. The rescattering of the struck nucleon debris
with the medium damps and distorts the momentum distributions of (A-1) in a way
which is very sensitive to the details of effective cross section of the
debris.
  The total cross section of the process on ^4He, ^16O, and ^40Ca, related to
the probability that after a target nucleon experiences a DIS process, the
recoiling (A-1) nucleus remains intact, is evaluated, and the distorted
momentum distribution of the recoiling (A-1) system is obtained. It is shown
that both quantities are very sensitive to the details of the early stage of
hadronization of the nucleon debris in the nuclear medium.
