  We study optimal Markovian couplings of Markov processes, where the
optimality is understood in terms of minimization of concave transport costs
between the time-marginal distributions of the coupled processes. We provide
explicit constructions of such optimal couplings for one-dimensional
finite-activity L\'evy processes (continuous-time random walks) whose jump
distributions are unimodal but not necessarily symmetric. Remarkably, the
optimal Markovian coupling does not depend on the specific concave transport
cost. To this end, we combine McCann's results on optimal transport and Rogers'
results on random walks with a novel uniformization construction that allows us
to characterize all Markovian couplings of finite-activity L\'evy processes. In
particular, we show that the optimal Markovian coupling for finite-activity
L\'evy processes with non-symmetric unimodal L\'evy measures has to allow for
non-simultaneous jumps of the two coupled processes.
