I am of school of thought that you should place stratum as a covariate in the model. possibility of interaction with stratum. If it is your covariates that define the strata then no alteration is necessary. Ive seen more complicated discussions on the matter, but this is my no nonsense approach.
It is because you are estimating
E(Y given covariates)
If the covariates define the strata, then it is the same as estimating the average in each strata, which is exactly what you are going to end up with anyway.