The OP is one such—Bayesians aren’t permitted to ignore any part of the data except those which leave the likelihood unchanged. One classic example is that in some problems, a confidence interval procedure can return the whole real line. A mildly less pathological example also concerning a wacky confidence interval is here.
Some frequentist techniques are strictly incoherent from a Bayesian point of view. In that case there is no prior.
I believe you and would like to know some examples for future reference.
The OP is one such—Bayesians aren’t permitted to ignore any part of the data except those which leave the likelihood unchanged. One classic example is that in some problems, a confidence interval procedure can return the whole real line. A mildly less pathological example also concerning a wacky confidence interval is here.