Semiclassical chiral fermion models with "Berry term" are studied in a symplectic framework. In the free case, the system proposed by Stephanov and Yin [1] can be obtained from Souriau's model for a relativistic massless spinning particle [2] by "enslaving the spin". The "Berry term" is identified with the classical spin two-form in [2]. The Souriau model carries a natural Poincare symmetry that we highlight, but boosts are broken by "spin enslavement". The relation between the models allows us to derive nevertheless a Poincare symmetry of an unconventional form for [1], which is shown to act differently from the natural one carried by the model [2]. Then we propose a general coupling procedure to an external electromagnetic field for any value of the gyromagnetic ratio $g$. For $g=0$ we get curious superluminal Hall-type motions, and for $g=2$ we recover spin precession. Our coupled theory is reminiscent of but different from other previously proposed ones.