  In this paper, we revisit a classic example of probabilistic cellular
automaton (PCA) on {0, 1} Z , namely, addition modulo 2 of the states of the
left-and right-neighbouring cells, followed by either preserving the result of
the addition, with probability p, or flipping it, with probability 1 -- p. It
is well-known that, for any value of p $\in$]0, 1[, this PCA is ergodic. We
show that, for p sufficiently close to 1, no coupling of the PCA dynamics based
on the composition of i.i.d. random functions of nearest-neighbour states (we
call this a basic i.i.d. coupling), can be successful, where successful means
that, for any given cell, the probability that every possible initial condition
leads to the same state after t time steps, goes to 1 as t goes to infinity. In
particular, this precludes the possibility of a CFTP scheme being based on such
a coupling. This property stands in sharp contrast with the case of monotone
PCA, for which, as soon as ergodicity holds, there exists a successful basic
i.i.d. coupling.
