Sampling among constrained partitions
A reflexive, symmetric and transitive relation $R$ on $O$ is the same as a partition of $O$. If $(o_i,o_j) \in R$ then $o_i,o_j$ are in the same partition, and so can be merged. If $(o_i,o_j) \notin R$ then $o_i,o_j$ are not allowed to be in the same partition. After merging, we reach the following problem:


  Given a graph, partition the set of vertices so that each partition is an independent set.


Equivalently (up to a permutation of colors):


  Given a graph, find a proper coloring.


Intuitively, it seems that a random partition will have many parts, so you can try to generate a random sample in the following way. Greedily sample a "small" independent set whose size you choose in advance as a Poisson random variable with constant expectation, remove it from the graph, and repeat. Of course, this won't be a uniform sample. You can gather statistics in some small cases to see what is a good choice of parameters in this procedure, and whether the results are good enough for your purposes.