  Higgs bundles are a central tool used to study a range of intersecting brane
systems in string compactifications. Solutions to the internal gauge theory
equations of motion for the corresponding worldvolume theories of branes give
rise to different low energy effective field theories. This has been heavily
used in the study of M-theory on local $G_2$ spaces and F-theory on local
elliptically fibered Calabi-Yau fourfolds. In this paper we show that the 3D
$\mathcal{N} = 1$ effective field theory defined by M-theory on a local
$Spin(7)$ space unifies the Higgs bundle data associated with 4D $\mathcal{N} =
1$ M- and F-theory vacua. This 3D system appears as an interface with finite
thickness between different 4D vacua. We develop the general formalism of
M-theory on such local $Spin(7)$ spaces, and build explicit interpolating
solutions. This provides a complementary local gauge theory analysis of a
recently proposed approach to constructing $Spin(7)$ spaces from generalized
connected sums.
