  We apply the BMHV scheme for non-anticommuting $\gamma_5$ to an abelian
chiral gauge theory at the two-loop level. As our main result, we determine the
full structure of symmetry-restoring counterterms up to the two-loop level.
These counterterms turn out to have the same structure as at the one-loop level
and a simple interpretation in terms of restoration of well-known Ward
identities. In addition, we show that the ultraviolet divergences cannot be
canceled completely by counterterms generated by field and parameter
renormalization, and we determine needed UV divergent evanescent counterterms.
The paper establishes the two-loop methodology based on the quantum action
principle and direct computations of Slavnov-Taylor identity breakings. The
same method will be applicable to nonabelian gauge theories.
