  We study fast-moving bright solitons in the focusing nonlinear Schr\"odinger
equation perturbed by a narrow Gaussian potential barrier. In particular, we
present a general and comprehensive analysis of the case where two fast-moving
bright solitons collide at the location of the barrier. In the limiting case of
a delta-function barrier, we use a quasi-analytic method to show that the
relative norms of the outgoing waves depends sinusoidally on the relative phase
of the incoming waves, and to determine whether one, or both, of the outgoing
waves are bright solitons. We show using numerical simulations that this
quasi-analytic result is valid in the high velocity limit: outside this limit
nonlinear effects introduce a skew to the phase-dependence, which we quantify.
Finally, we numerically explore the effects of introducing a finite-width
Gaussian barrier. Our results are particularly relevant, as they can be used to
describe a range of interferometry experiments using bright solitary
matter-waves.
