I meant stratified resampling algorithm by Kitagawa (1996) as implemented, e.g., in posterior package function resample_draws(). It reduces the variance in the resampling.
I had not thought about making the resampling independently for each chain, but that makes perfect sense so that Rhat is still available. ESS will be slightly wrong as the repeated draws will partially break the autocorrelation structure, but when importance sampling starts to fail Rhat part inside ESS computation dominates anyway.