I wouldn't say "no Bayesian explanation," but perhaps "a Bayesian explanation is unknown to me, nor do I see how this explanation would illuminate anything." But yes, I gave an example elsewhere in this thread. The FCI algorithm for learning graph structure in the non-parametric setting with continuous valued variables, where the correct underlying model has the following independence structure:

A is independent of B and C is independent of D (and nothing else is true).

Since I (and to my knowledge everyone else) do not know how to write the likelihood for this model, I don't know how to set up the standard Bayesian story here.