  We study two well-known $SU(N)$ chiral gauge theories with fermions in the
symmetric, anti-symmetric and fundamental representations. We give a detailed
description of the global symmetry, including various discrete quotients.
Recent work argues that these theories exhibit a subtle mod 2 anomaly, ruling
out certain phases in which the theories confine without breaking their global
symmetry, leaving a gapless composite fermion in the infra-red. We point out
that no such anomaly exists. We further exhibit an explicit path to the gapless
fermion phase, showing that there is no kinematic obstruction to realising
these phases.
