  We develop a hybrid semiclassical method to study the time evolution of one
dimensional quantum systems in and out of equilibrium. Our method handles
internal degrees of freedom completely quantum mechanically by a modified time
evolving block decimation method, while treating orbital quasiparticle motion
classically. We can follow dynamics up to timescales well beyond the reach of
standard numerical methods to observe the crossover between pre-equilibrated
and locally phase equilibrated states. As an application, we investigate the
quench dynamics and phase fluctuations of a pair of tunnel coupled one
dimensional Bose condensates. We demonstrate the emergence of soliton-collision
induced phase propagation, soliton-entropy production and multistep
thermalization. Our method can be applied to a wide range of gapped
one-dimensional systems.
