Is post-stratification inherently non-Bayesian? It is increasingly common to employ regression with post-stratification. Since probability-weighting is incoherent in Bayesian inference (thus why sampling/survey weights and weighted psuedo-likelihoods are not used), post-stratification is used to infer treatment effects for populations known to differ from one's sample from a model with the available data.
The gist is that you specify a model where a parameter (e.g. a treatment indicator) varies across other parameters (e.g. a multiplicative interaction with demographic indicators), calculate the expectation for each combination of parameter values, then get a "population-level" expectation by taking a weighted average of combination expectations with weights proportional to the known prevalence of those combinations in some population.
Since this method assumes a "population" distribution of parameter values, is this inherently non-Bayesian?

No. I don't know why the use of population cell proportions would be considered non-Bayesian. The key distinction between Bayesian and non-Bayesian methods is that in Bayesian methods, we use probability to model uncertainty about components of a model. Post-stratification simply introduces additional components to the model (i.e., population cell sizes), which we can treat as either known with certainty or as unknown (in which case we model them with priors and likelihoods).
Typically, the population cell proportions are assumed known. But that assumption can be weakened by modeling the cell proportions and using a non-degenerate prior distribution. For example, the "Bayes-raking" method  models cell proportions using information about marginal population totals and models for the sample selection/nonresponse mechanisms.
Whether we treat the cell proportions as known or unknown, at the end of the day we still use prior distributions for some components of the model and we still ultimately use posterior distributions to make inferences. So as far as I can see, there's nothing inherently non-Bayesian about post-stratification.
