Consider sampling from an unknown probability distribution on the integers. With a tail-free prior, the posterior distribution is consistent. With a mixture of a tail-free prior and a point mass, however, the posterior may be inconsistent. This is likewise true for a countable mixture of tail-free priors. Similar results are given for Dirichlet priors.
"On Inconsistent Bayes Estimates in the Discrete Case." Ann. Statist. 11 (4) 1109 - 1118, December, 1983. https://doi.org/10.1214/aos/1176346325