  We examine the discontinuous first-order superfluid $^3$He A to B transition
in the vicinity of the polycritical point (2.232 mK and 21.22 bar). We find
path-dependent transitions: cooling at fixed pressure yields a well defined
transition line in the temperature-pressure plane, but this line can be
reliably crossed by depressurizing at nearly constant temperature after
transiting $T_{\rm c}$ at a higher pressure. This path dependence is not
consistent with any of the standard B-phase nucleation mechanisms in the
literature. This symmetry breaking transition is a potential simulator for
first order transitions in the early universe.
