Mixture models pose all sorts of identifiability problems.
This post exhibits a subtle one with just gaussians. If the means are close and the variances high enough, there’s nearly no way to figure out how many gaussians there should be. You can put “repulsive” priors to help a bit but then you need some “attractive” priors too! It just gets really unwieldy, even in a simple Gaussian mixture model.
See also @betanalpha Identifying Bayesian Mixture Models (betanalpha.github.io).