  We extend the construction of lattice chiral gauge theories based on
non-perturbative gauge fixing to the non-abelian case. A key ingredient is that
fermion doublers can be avoided at a novel type of critical point which is only
accessible through gauge fixing, as we have shown before in the abelian case.
The new ingredient allowing us to deal with the non-abelian case as well is the
use of equivariant gauge fixing, which handles Gribov copies correctly, and
avoids Neuberger's no-go theorem. We use this method in order to gauge fix the
non-abelian group (which we will take to be SU(N)) down to its maximal abelian
subgroup. Obtaining an undoubled, chiral fermion content requires us to
gauge-fix also the remaining abelian gauge symmetry. This modifies the
equivariant BRST identities, but their use in proving unitarity remains intact,
as we show in perturbation theory. On the lattice, equivariant BRST symmetry as
well as the abelian gauge invariance are broken, and a judiciously chosen
irrelevant term must be added to the lattice gauge-fixing action in order to
have access to the desired critical point in the phase diagram. We argue that
gauge invariance is restored in the continuum limit by adjusting a finite
number of counter terms. We emphasize that weak-coupling perturbation theory
applies at the critical point which defines the continuum limit of our lattice
chiral gauge theory.
