  Compactification of M- / string theory on manifolds with $G_2$ structure
yields a wide variety of 4D and 3D physical theories. We analyze the local
geometry of such compactifications as captured by a gauge theory obtained from
a three-manifold of ADE singularities. Generic gauge theory solutions include a
non-trivial gauge field flux as well as normal deformations to the
three-manifold captured by non-commuting matrix coordinates, a signal of
T-brane phenomena. Solutions of the 3D gauge theory on a three-manifold are
given by a deformation of the Hitchin system on a marked Riemann surface which
is fibered over an interval. We present explicit examples of such backgrounds
as well as the profile of the corresponding zero modes for localized chiral
matter. We also provide a purely algebraic prescription for characterizing
localized matter for such T-brane configurations. The geometric interpretation
of this gauge theory description provides a generalization of twisted connected
sums with codimension seven singularities at localized regions of the geometry.
It also indicates that geometric codimension six singularities can sometimes
support 4D chiral matter due to physical structure "hidden" in T-branes.
