Dear @damarginal,
very interesting question. Indeed, modeling the dependence among the parameters of a model - any, therefore also a model obtained by Bayesian methods - makes a lot of sense to me. For instance, it enables a more accurate resampling from that model.
We had a similar case to yours - we were using parametric models of earthquake signals across Europe given historical data, and found strong correlations between some parameters in the model. So we used copulas to model such dependence. We even observed strong tail correlations, hinting at the fact that the parameter models tended to exhibit joint extreme values considerably more often than if they were statistically independent. This, in turn, totally changes the statistics of the signals and the shape of the hazard functions.
From an information theoretical point of view, we could say that these models have some degree of redundancy in their parameters: given the value of some parameters, other parameters are very likely to be confined into a smaller range of likely values, and unlikely to deviate much from those. Unfortunately, I have not found any publication specifically addressing this point. If you find any, please leave a note!