I know that multiple imputation requires you to include at least every variable in your final estimation model (including interactions) and variables not included but later added to the dataset will have their "relationship with other variables in the dataset biased (normally towards zero)."

However, what happens when you include a variable in the dataset during imputation and then use that variable to stratify the regressions (as with the "by variname:" option in Stata)? Effectively, this stratification mimics an interaction by allowing coefficient values to vary across regressions within the sample; are the results of this kind of procedure still BLUE (providing other regression presumptions are met).