  We elaborate the formulation of the $\mathsf{CP^{n-1}}$ sigma model with
fermions as a gauged Gross-Neveu model. This approach allows to identify the
super phase space of the model as a supersymplectic quotient. Potential chiral
gauge anomalies are shown to receive contributions from bosons and fermions
alike and are related to properties of this phase space. Along the way we
demonstrate that the worldsheet supersymmetric model is a supersymplectic
quotient of a model with target space supersymmetry. Possible generalizations
to other quiver supervarieties are briefly discussed.
