  We show that the stabilization of any countable ergodic p.m.p. equivalence
relation which is not Schmidt, i.e. admits no central sequences in its full
group, always gives rise to a stable equivalence relation with a unique stable
decomposition, providing the first non-strongly ergodic such examples. In the
proof, we moreover establish a new local characterization of the Schmidt
property. We also prove some new structural results for product equivalence
relations and orbit equivalence relations of diagonal product actions.
