  We study interactions between bright matter-wave solitons which acquire
chiral transport dynamics due to an optically-induced density-dependent gauge
potential. Through numerical simulations, we find that the collision dynamics
feature several non-integrable phenomena, from inelastic collisions including
population transfer and radiation losses to short-lived bound states and
soliton fission. An effective quasi-particle model for the interaction between
the solitons is derived by means of a variational approximation, which
demonstrates that the inelastic nature of the collision arises from a coupling
of the gauge field to velocities of the solitons. In addition, we derive a set
of interaction potentials which show that the influence of the gauge field
appears as a short-range potential, that can give rise to both attractive and
repulsive interactions.
