  For a NIP theory $T$, a sufficiently saturated model $\mathfrak{C}$ of $T$,
and an invariant (over some small subset of $\mathfrak{C}$) global type $p$, we
prove that there exists a finest relatively type-definable over a small set of
parameters from $\mathfrak{C}$ equivalence relation on the set of realizations
of $p$ which has stable quotient. This is a counterpart for equivalence
relations of the main result of the paper "On maximal stable quotients of
definable groups in NIP theories" by M. Haskel and A. Pillay which shows the
existence of maximal stable quotients of type-definable groups in NIP theories.
Our proof adapts the ideas of the proof of this result, working with relatively
type-definable subsets of the group of automorphisms of the monster model as
defined in the paper "On first order amenability" by E. Hrushovski, K.
Krupinski, and A. Pillay.
