  Choosing how gauge $U(1)_\chi$ breaks in the context of $SO(10) \to SU(5)
\times U(1)_\chi$, $Z_4$ lepton number may be obtained which maintains
neutrinos as Dirac fermions. Choosing $\Delta(27)$ as the family symmetry of
leptons, tree-level Dirac neutrino masses may be forbidden. Choosing a specific
set of self-interacting dark-matter particles, Dirac neutrino masses and mixing
may then be generated in two loops. This framework allows the realization of
cobimaximal neutrino mixing, i.e. $\theta_{13} \neq 0$, $\theta_{23} = \pi/4$,
$\delta_{CP} = \pm \pi/2$, as well as the desirable feature that the light
scalar mediator of dark-matter interactions decays only to neutrinos, thereby
not disrupting the cosmic microwave background (CMB).
