Abstract. The Last Glacial Maximum (LGM) was characterised by huge ice sheets covering the Northern Hemisphere, especially over North America, and by its cold climate. Previous authors have performed numerical simulations of the LGM to better understand coupled climate–ice sheet systems. However, the results of such simulations are sensitive to many model parameters. Here, we perform a 200-member ensemble of simulations of the North American and Greenland ice sheets and climate of the LGM with a coupled ice sheet–atmosphere–slab ocean model (FAMOUS-BISICLES) to explore sensitivities of the coupled climate–ice system to 16 uncertain parameters. In the ensemble of simulations, the global mean surface temperature is primarily controlled by the combination of parameters in the large-scale condensation scheme and the cumulus convection scheme. In simulations with plausible LGM global mean surface temperatures, we find that the albedo parameters have only a small impact on the Greenland ice volume due to the limited area of surface ablation associated with the cold climate. Instead, the basal sliding law controls the ice volume by affecting ice transport from the interior to the margin. On the other hand, like the Greenland ice sheet in future climate change, the LGM North American ice sheet volume is controlled by parameters in the snow and ice albedo scheme. Few of our simulations produce an extensive North American ice sheet when the global temperature is above 12 °C. Based on constraints on the LGM global mean surface temperature, the ice volume and the southern extent of the North American ice sheet, we select 16 acceptable simulations. These simulations lack the southern extent of ice compared to reconstructions, but they show reasonable performance on the ice sheet configuration and ice streams facing Baffin Bay and the Arctic Ocean. The strong sensitivities of the North American ice sheet to albedo at the LGM may imply a potential constraint on the future Greenland ice sheet by constraining the albedo schemes.