Models of random spatial permutations arise in the study of Bose-Einstein condensation. Namely, permutations of sites occur with probabilities depending on lengths of permutation jumps, as well as on interactions between jumps. Below a critical temperature, one observes the onset of long permutation cycles in spite of short-distance permutation-jump interactions. We have studied this model using MCMC methods. Our current best algorithm, however, only produces permutations with even winding numbers on the 3-torus. Similar problems have been observed, and resolved, in PIMC studies via worm algorithms. Our worm algorithm, inspired by the PIMC approach, produces winding numbers of both parities, but suffers from a stopping-time involving worm closures; attempts to remedy this fail to satisfy detailed balance. Insight into this problem is welcome.