  We study non-BPS D-branes in the type II string vacua with chiral space-time
SUSY constructed based on the asymmetric orbifolds of $\mbox{K3} \cong
T^4/{\Bbb Z}_2$ as a succeeding work of arXiv:1704.05262[hep-th]. We especially
focus on the fractional D-branes that contains the contributions from the
twisted sector of the ${\Bbb Z}_2$ orbifolding.
  We show that the cylinder partition functions for these fractional branes do
not vanish, as in the cases of ordinary non-BPS D-branes. This aspect is in a
sharp contrast with the bulk-type branes studied in arXiv:1704.05262[hep-th].
We then discuss the extensions of models by including the discrete torsion
depending on the spin structures, and investigate whether we obtain the
vanishing self-overlaps associated to the fractional branes. The
existence/absence of bose-fermi cancellations both in the closed and open
string spectra as well as the massless spectra in the twisted sectors crucially
depend on the discrete torsion. We find that some choices of the torsion indeed
realize the vanishing self-overlaps for the fractional branes, with keeping the
vanishing torus partition function intact.
