  Semiclassical chiral fermion models with Berry term are studied in a
symplectic framework. In the free case, the system can be obtained from
Souriau's model for a relativistic massless spinning particle by "enslaving"
the spin. The Berry term is identified with the classical spin two-form of the
latter model. The Souriau model carries a natural Poincar\'e symmetry that we
highlight, but spin enslavement breaks the boost symmetry. However the relation
between the models allows us to derive a Poincare symmetry of unconventional
form for chiral fermions. Then we couple our system to an external
electromagnetic field. For gyromagnetic ratio $g=0$ we get curious superluminal
Hall-type motions; for $g=2$ and in a pure constant magnetic field in
particular, we find instead spiraling motions.
