A pressing challenge in modern survey research is to find calibration weights when covariates are high dimensional and especially when interactions between variables are important. Traditional approaches like raking typically fail to balance higher-order interactions; and post-stratification, which exactly balances all interactions, is only feasible for a small number of variables. In this paper, we propose multilevel calibration weighting, which enforces tight balance constraints for marginal balance and looser constraints for higher-order interactions. This incorporates some of the benefits of post-stratification while retaining the guarantees of raking. We then correct for the bias due to the relaxed constraints via a flexible outcome model; we call this approach Double Regression with Post-stratification (DRP). We characterize the asymptotic properties of these estimators and show that the proposed calibration approach has a dual representation as a multilevel model for survey response. We assess the performance of this method via an extensive simulation study and show how it can reduce bias in a case-study of a large-scale survey of voter intention in the 2016 U.S. presidential election. The approach is available in the multical R package.