General real-valued theories with the Schröder-Bernstein property are stable

We show that every general theory à la Keisler with the Schröder-Bernstein property is stable. This generalizes the corresponding result from classical logic due to John Goodrick. Our proof uses the classical result (generalized to the case that the instability is witnessed by an infinitary formula) together with a discretization technique introduced by Keisler and the third-named author. We speculate on how our techniques could be adapted to show that every continuous theory with the Schröder-Bernstein property is stable.