We present a novel mapmaking method for obtaining unbiased estimates of CMB polarisation, tailored to modern CMB experiments with a rotating half-wave plate. These experiments are exposed to strong unpolarised contaminant sources, such as atmospheric emission and ground pickup, which can be several orders of magnitude stronger than the sky signal. Our mapmaker mitigates these systematic effects by marginalising over all signals that vary slowly compared to the timescale of a polarimeter’s angle rotation on the sky, while recovering high-fidelity polarisation maps. When the variability timescales of the unpolarised signals exceed a quarter of the half-wave plate rotation period, the method can produce maps with nearly optimal noise levels and minimal contamination. Furthermore, if the half-wave plate rotation period is sufficiently short relative to the beam-scale crossing time, the method efficiently mitigates the sky intensity-to-polarisation leakage. In this work we demonstrate these features using simulations including sky signal, instrumental noise, and atmospheric emissions. This mapmaker, named the Polarisation-Optimised Map-Making Estimator (POMME), is implemented within the open-source FURAX package and is ready for application to upcoming ground-based CMB surveys.