  Stein discrepancies have emerged as a powerful tool for retrospective
improvement of Markov chain Monte Carlo output. However, the question of how to
design Markov chains that are well-suited to such post-processing has yet to be
addressed. This paper studies Stein importance sampling, in which weights are
assigned to the states visited by a $\Pi$-invariant Markov chain to obtain a
consistent approximation of $P$, the intended target. Surprisingly, the optimal
choice of $\Pi$ is not identical to the target $P$; we therefore propose an
explicit construction for $\Pi$ based on a novel variational argument. Explicit
conditions for convergence of Stein $\Pi$-Importance Sampling are established.
For $\approx 70\%$ of tasks in the PosteriorDB benchmark, a significant
improvement over the analogous post-processing of $P$-invariant Markov chains
is reported.
