  We study the large $N$ saddle points of two SYK chains coupled by an
interaction that is nonlocal in Euclidean time. We start from analytic
treatment of the free case with $q=2$ and perform the numerical study of the
interacting case $q=4$. We show that in both cases there is a nontrivial phase
structure with infinite number of phases. Every phase correspond to a saddle
point in the non-interacting two-replica SYK. The nontrivial saddle points have
non-zero value of the replica-nondiagonal correlator in the sense of
quasi-averaging, when the coupling between replicas is turned off. Thus, the
nonlocal interaction between replicas provides a protocol for turning the
nonperturbatively subleading effects in SYK into non-equilibrium configurations
which dominate at large $N$. For comparison we also study two SYK chains with
local interaction for $q=2$ and $q=4$. We show that the $q=2$ model also has a
similar phase structure, whereas in the $q=4$ model, dual to the traversable
wormhole, the phase structure is different.
