Motivated by privacy constraints which force to unlabel multiple samples from a discrete distribution, we consider the problem of dealing with a Bayesian model when conditioning on multiple partitions induced by these samples. Aiming at evaluating the distribution of coagulations produced by matching the groups of the multiple partitions, whose original type is unknown, and motivated by the high computational cost of exact evaluations, we formulate a Metropolis–Hastings sampler that is shown to yield good approximations in reasonable computing time despite the great sparsity displayed by the target distribution.