We consider Markov chains {Γ n } with transitions of the form Γ n =f(X n ,Y n )Γ n - 1 +g(X n ,Y n ), where {X n } and {Y n } are two independent i.i.d. sequences. For two copies {Γ n } and {Γ n '} of such a chain, it is well known that L(Γ n )-L(Γ n ') 0 provided E[log(f(X n ,Y n ))]<0, where is weak convergence. In this paper, we consider chains for which also ||Γ n -Γ n '||->0, where ||.|| is total variation distance. We consider in particular how to obtain sharp quantitative bounds on the total variation distance. Our method involves a new coupling construction, one-shot coupling, which waits until time n before attempting to couple. We apply our results to an auto-regressive Gibbs sampler, and to a Markov chain on the means of Dirichlet processes.