  We consider $SU(N)$ $\mathcal{N}=4$ super Yang-Mills with cubic and quadratic
soft SUSY breaking potential, such that the global $SU(4)_R$ is broken to
$SU(3)$ or further. As shown recently, this set-up supports a rich set of
non-trivial vacua with the geometry of self-intersecting $SU(3)$ branes in 6
extra dimensions. The zero modes on these branes can be interpreted as 3
generations of bosonic and chiral fermionic strings connecting the branes at
their intersections. Here, we uncover a large class of exact solutions
consisting of branes connected by Higgs condensates, leading to Yukawa
couplings between the chiral fermionic zero modes. Under certain decoupling
conditions, the backreaction of the Higgs on the branes vanishes exactly. The
resulting physics is that of a spontaneously broken chiral gauge theory on
branes with fluxes. In particular, we identify combined brane plus Higgs
configurations which lead to gauge fields that couple to chiral fermions at low
energy. This turns out to be quite close to the Standard Model and its
constructions via branes in string theory. As a by-product, we construct a
$G_2$-brane solution corresponding to a squashed fuzzy coadjoint orbit of
$G_2$.
