Strategy for adding more model parameters within the blocked Gibbs Sampling framework by adding more blocks

I had an idea while watching a video on Gibbs sampling (the bit beginning at roughly 1h09m): Maybe we can adapt the Gibbs sampler to deal with correlated parameters after all. I'm not sure how to implement it, but the main procedure would be something like this:

1. Update ~6 Nuissance Parameters, holding everything else fixed
2. Update ~6 traditional stellar parameters (à-la Czekala et al. 2015)
3. new Update whatever new stellar model parameters you've added (e.g. veiling, starspots, binary, whatever)

My guess is that this three-level blocked Gibbs sampler would outperform a two-level Gibbs with the new stellar parameters included in block 2.

For example, I witnessed poor convergence when I attempted to fit a mixture model with ~8 stellar parameters (3 of which were strongly correlated) in block 2, as discussed in detail in Issue #35. That tension led me to a major departure from the Gibbs sampler, in which I run emcee to sample all 14 stellar+nuisance parameters simultaneously, but with the major limitation that I had to chunk by spectral order thereby deriving multiple independent sets of stellar posteriors for ~50+ spectral orders. The extension to the Gibbs framework described here, if it can be implemented, would return to the much better situation of having a single posterior that is consistent with all the data.

This seems obvious in hindsight, so it must be a good idea, right?

Start by reviewing the existing blocked Gibbs sampler and Issue #35, then compare the proposed three-block strategy with the documented emcee approach. The issue does not name files, tests, or an implementation entry point; done would require an agreed design and evidence that the revised sampler improves convergence while producing one posterior across all data.