  Some models of asymmetric dark matter commonly employ a gauge group structure
of the form $G_{V}\times{}G_{D}$ where $G_{V}$ is the visible gauge group
containing the Standard Model and $G_{D}$ is the gauge group responsible for
self-interactions amongst components of dark matter. In some models, there is
also an additional spontaneously broken $U(1)$ gauge symmetry coupling the
visible and dark sectors at high energies. One theoretical problem is how to
unify the visible and dark sectors by inducing the spontaneous breaking
$G\rightarrow{}G_{V}\times{}G_{D}$ for some large gauge group $G$. In this
paper, we discuss how to generate such a structure at low energies, in the
context of 4+1-dimensional domain-wall brane model, by employing a
generalization of the Dvali-Shifman mechanism, used to localized gauge bosons
on domain walls, called the clash-of-symmetries mechanism. In one model, we
describe a clash-of-symmetries domain wall solution in a theory with two scalar
fields in the adjoint representation which breaks the group $SU(12)$ to two
differently embedded copies of $SU(6)\times{}SU(6)\times{}U(1)$, leading to a
an effective $SU(5)_{V}\times{}SU(5)_{D}\times{}U(1)_{X}$-invariant field
theory on the wall. We find that fermions in the mixed representations of
$SU(5)_{V}\times{}SU(5)_{D}$ do not couple to the domain wall and thus remain
5D vector-like Dirac fermions, attaining masses of order $M_{GUT}$ when we
perform the breaking
$SU(5)_{V}\rightarrow{}SU(3)_{c}\times{}SU(2)_{I}\times{}U(1)_{Y}$, thus being
removed from the spectrum. We also outline how to build a few alternative
models, one based on the group $SU(9)$, and a couple more based on
non-clash-of-symmetries domain wall solutions in $SU(12)$ and $SU(10)$ models.
