  Bidirectional transport in (quasi) one-dimensional systems generically leads
to cluster-formation and small particle currents. This kind of transport can be
described by the asymmetric simple exclusion process (ASEP) with two species of
particles. In this work, we consider the effect of non-Markovian site exchange
times between particles. Different realizations of the exchange process can be
considered: The exchange times can be assigned to the lattice bonds or each
particle. In the latter case we specify additionally which of the two exchange
times is executed, the earlier one (minimum rule) or the later one (maximum
rule). In a combined numerical and analytical approach we find evidence that we
recover the same asymptotic behavior as for unidirectional transport for most
realizations of the exchange process. Differences in the asymptotic behavior of
the system have been found for the minimum rule which is more efficient for
fast decaying exchange time distributions.
