  We study the classical cubic-lattice double dimer model, consisting of two
coupled replicas of the close-packed dimer model, using a combination of
theoretical arguments and Monte Carlo simulations. Our results establish the
presence of a 'synchronization transition' at a critical value of the coupling,
where both replicas remain disordered but their fluctuations become strongly
correlated. We show that this unconventional transition, which has neither
external nor spontaneous symmetry breaking, is continuous and belongs to the 3D
inverted-XY universality class. By adding aligning interactions for dimers
within each replica, we map out the full phase diagram including the interplay
between columnar ordering and synchronization. We also solve the coupled double
dimer model exactly on the Bethe lattice and show that it correctly reproduces
the qualitative phase structure, but with mean-field critical behavior.
