  We explore a transverse-field Ising model that exhibits both spontaneous
symmetry-breaking and eigenstate thermalization. Within its ferromagnetic
phase, the exact eigenstates of the Hamiltonian of any large but finite-sized
system are all Schr\"odinger cat states: superpositions of states with `up' and
`down' spontaneous magnetization. This model exhibits two dynamical phase
transitions {\it within} its ferromagnetic phase: In the lowest-temperature
phase the magnetization can macroscopically oscillate between up and down. The
relaxation of the magnetization is always overdamped in the remainder of the
ferromagnetic phase, which is divided in to phases where the system thermally
activates itself {\it over} the barrier between the up and down states, and
where it quantum tunnels.
