We knew that in a likelihood framework, the no common mechanism model's ML tree was the same as the maximum parsimony tree. It is not really surprising, given the commonalities between likelihood and Bayesian frameworks, that this equivalence holds for Huelsenbeck et al.'s new implementation (see their figure 5, above, showing a perfect negative relationship between log-likelihood and parsimony score). Still, there are a number of very interesting tidbits in this paper. First, one can now compare the "fit" of a parsimony model to other, more commonly used, Bayesian models. Second, this framework provides natural measures of support for a parsimony analysis, rather than approximations like bootstrap values. The catch here is that the paper itself provides compelling arguments against even implementing this model:
The no-common mechanism model is very peculiar, and the authors have mixed feelings about having implemented the method in a Bayesian framework. (Huelsenbeck et al., p. 415)
There's also a couple nifty new tree-searching "tricks" that the authors implement to help search tree space more efficiently under this complex model. These two (Gibbs-like TBR move and Gibbs eraser) may affect your life, some day, by making your tree searches run faster.