The coupling-from-the-past (CFTP) algorithm of Propp and Wilson, also called perfect sampling, permits one to sample exactly from the stationary distribution of an ergodic Markov chain. By using it n times independently, we obtain an independent sample from that distribution. A more representative sample can be obtained by creating negative dependence between these n replicates; other authors have already proposed to do this via antithetic variates, Latin hypercube sampling, and randomized quasi-Monte Carlo (RQMC). We study a new, often more effective, way of combining CFTP with RQMC, based on the array-RQMC algorithm. We provide numerical illustrations for Markov chains with both finite and continuous state spaces, and compare with the RQMC combinations proposed earlier.