  We study a two-site Sachdev-Ye-Kitaev model with complex couplings and a weak
inter-site interaction. At low temperatures, the system is dual to a Euclidean
wormhole in Jackiw-Teitelboim gravity plus matter. Interestingly, the energy
spectrum becomes real for sufficiently strong inter-site coupling despite the
Hamiltonian being non-Hermitian. In gravity, this complex-to-real transition
corresponds to a Euclidean-to-Lorentzian transition: a dynamical restoration of
the gravitational SL(2,R) symmetry of the Lorentzian wormhole, broken to U(1)
in the Euclidean wormhole. We show this by identifying an order parameter for
the symmetry breaking and by matching the oscillating patterns of the Green's
functions. Above the transition, the system can be continued to Lorentzian
signature and is dual to an eternal traversable wormhole. Additionally, we
observe a thermal phase transition from the wormhole to two black holes and
provide a detailed matching of the associated physical quantities. The analysis
of level statistics reveals that in a broad range of parameters the dynamics is
quantum chaotic in the universality class of systems with time reversal
invariance.
