  We explicitly construct a coupling attaining Ornstein's $\bar{d}$-distance
between ordered pairs of binary chains of infinite order. Our main tool is a
representation of the transition probabilities of the coupled bivariate chain
of infinite order as a countable mixture of Markov transition probabilities of
increasing order. Under suitable conditions on the loss of memory of the
chains, this representation implies that the coupled chain can be represented
as a concatenation of iid sequence of bivariate finite random strings of
symbols. The perfect simulation algorithm is based on the fact that we can
identify the first regeneration point to the left of the origin almost surely.
