  One of the interesting features in unification models and supersymmetric
unification models is that the chiral states of quarks and leptons in a family
including a right-handed neutrino can be fitted neatly into a fundamental
spinor representation (f.s) of dimension 16 for the SO(10) gauge group.
However, it is shown in this paper that such a fundamental spinor
representation of SO(10) for Weyl fermions will generate global
(non-perturbative) gauge anomalies (of new type) when restricting to the
$SU(2)\otimes SU(2)\otimes SU(2) \otimes SU(2)$ gauge subgroup. Such an example
is the four SU(2) factors obtained through the reduction of the subgroup
$SU(2)\otimes SO(7)$ of SO(10) with the SO(7) to the three SU(2) factors. The
branching rule in this case is given by $(f.s)\rightarrow (2-1-2-2)\oplus
(2-2-1-2)$ in terms of dimensions. A consistent gauge theory implies the gauge
symmetry in a gauge subgroup, and then needs to be well-defined when
restricting to the gauge subgroup. Consequently, a consistent SO(10) quantum
theory needs to satisfy our selection rule $N_f+N_{mf}=even\geq 4$, namely the
total number of generations with $N_f$ ordinary fermion families and $N_{mf}$
mirror fermion families is even and larger than three, and the three
generations of chiral fermions in this content can not correspond to a
consistent theory. Then we expect that there exist at least one additional
fermion family including a right-handed neutrino or at least one family of
mirror fermions including a left-handed mirror neutrino if SO(10) unification
theory is relevant to our realistic world. The next possibility is to have
total six generations that may subject to stringent
