  Dealing with finite Markov chains in discrete time, the focus often lies on
convergence behavior and one tries to make different copies of the chain meet
as fast as possible and then stick together. There is, however, a very peculiar
kind of discrete finite Markov chain, for which two copies started in different
states can be coupled to meet almost surely in finite time, yet their
distributions keep a total variation distance bounded away from 0, even in the
limit as time goes off to infinity. We show that the supremum of total
variation distance kept in this context is $\frac12$.
