  The entanglement between two weakly coupled bosonic Josephson junctions is
studied in relation to the classical mixed phasespace structure of the system,
containing symmetry-related regular islands separated by chaos. The
symmetry-resolved entanglement spectrum and bi-partite entanglement entropy of
the system's energy eigenstates are calculated and compared to their expected
structure for random states that exhibit complete or partial ergodicity. The
entanglement spectra of chaos-supported eigenstates match the microcanonical
structure of a Generalized Gibbs Ensemble due to the existence of an adiabatic
invariant that restricts ergodization on the energy shell. The
symmetry-resolved entanglement entropy of these quasistochastic states consists
of a mean-field maximum entanglement term and a fluctuation correction due to
the finite size of the constituent subsystems. The total bi-partite
entanglement entropy of the eigenstates correlates with their chaoticity.
Island-supported eigenstates are macroscopic Schr\"odinger cat states for
particles and excitations, with substantially lower entanglement.
