  We entertain the possibility that neutrino masses and dark matter (DM)
originate from a common composite dark sector. A minimal effective theory can
be constructed based on a dark $SU(3)_D$ interaction with three flavors of
massless dark quarks; electroweak symmetry breaking gives masses to the dark
quarks. By assigning a $\mathbb Z_2$ charge to one flavor, a stable "dark kaon"
can provide a good thermal relic DM candidate. We find that "dark neutrons" may
be identified as right handed Dirac neutrinos. Some level of
"neutron-anti-neutron" oscillation in the dark sector can then result in
non-zero Majorana masses for light Standard Model neutrinos. A simple
ultraviolet completion is presented, involving additional heavy
$SU(3)_D$-charged particles with electroweak and lepton Yukawa couplings. At
our benchmark point, there are "dark pions" that are much lighter than the
Higgs and we expect spectacular collider signals arising from the UV framework.
This includes the decay of the Higgs boson to $\tau \tau \ell \ell^{\prime}$,
where $\ell$($\ell'$) can be any lepton, with displaced vertices. We discuss
the observational signatures of this UV framework in dark matter searches and
primordial gravitational wave experiments; the latter signature is potentially
correlated with the $H \to \tau \tau \ell \ell^{\prime}$ decay.
