  We study Floquet engineering of the tunnel coupling between a pair of
one-dimensional bosonic quasi-condensates in a tilted double-well potential. By
modulating the energy difference between the two wells, we re-establish tunnel
coupling and precisely control its amplitude and phase. This allows us to
initiate coherence between two initially uncorrelated Bose gases and prepare
different initial states in the emerging sine-Gordon Hamiltonian. We fully
characterize the Floquet system and study the dependence of both equilibrium
properties and relaxation on the modulation.
