  Ricci flat manifolds of special holonomy are a rich framework as models of
the extra dimensions in string/$M$-theory. At special points in vacuum moduli
space, special kinds of singularities occur and demand a physical
interpretation. In this paper we show that the topologically distinct
$G_2$-holonomy manifolds arising from desingularisations of codimension four
orbifold singularities due to Joyce and Karigiannis correspond physically to
Coulomb and Higgs phases of four dimensional gauge theories. The results
suggest generalisations of the Joyce-Karigiannis construction to arbitrary
ADE-singularities and higher order twists which we explore in detail in
explicitly solvable local models. These models allow us to derive an
isomorphism between moduli spaces of Ricci flat metrics on these non-compact
$G_2$-manifolds and flat ADE-connections on compact flat 3-manifolds which we
establish explicitly for $\operatorname{SU}(n)$.
