You may be right about that. I suspected that the approach that I was thinking about would be naive.
Anyway, I read further in that paper by Plummer and noticed this:
The target distribution p∗ (θ ) can always be estimated by multiple imputation (MI), i.e. generating a sequence of samples φ(1) , . . . φ(T) and then fitting T separate models for θ such that, under model t, φ is considered to be observed at φ = φ(t). Pooled MCMC samples from all T models can be used to estimate the marginal density p∗ (θ ).
That looks like it may be feasible for me, even if it’s rather a brute force approach. Would I be correct in gathering that pooling MCMC samples to get a posterior distribution would pretty much be a matter of concatenating or shuffling together all the chains from the MCMC simulations and then generating a histogram or kernel density estimation from the samples from all of the chains?