  The stability of hybrid stars with first-order phase transitions as
determined by calculating fundamental radial oscillation modes is known to
differ from the predictions of the widely-used Bardeen--Thorne--Meltzer
criterion. We consider the effects of out-of-chemical-equilibrium physics on
the radial modes and hence stability of these objects. For a barotropic
equation of state, this is done by allowing the adiabatic sound speed to differ
from the equilibrium sound speed. We show that doing so extends the stable
branches of stellar models, allowing stars with rapid phase transitions to
support stable higher-order stellar multiplets similarly to stars with multiple
slow phase transitions. We also derive a new junction condition to impose on
the oscillation modes at the phase transition. Termed the reactive condition,
it is physically motivated, consistent with the generalized junction conditions
between two phases, and has the common rapid and slow conditions as limiting
cases. Unlike the two common cases, it can only be applied to nonbarotropic
stars. We apply this junction condition to hybrid stellar models generated
using a two-phase equation of state consisting of nuclear matter with unpaired
quark matter at high densities joined by a first-order phase transition and
show that like in the slow limiting case, stars that are classically unstable
are stabilized by a finite chemical reaction speed.
