The evolution of past global ice sheets is highly uncertain. One example is the missing ice problem during the Last Glacial Maximum (LGM, 26 000-19 000 years before present) – an apparent 8-28 m discrepancy between far-field sea level indicators and modelled sea level from ice sheet reconstructions.In the absence of ice sheet reconstructions, researchers often use marine δ¹⁸O proxy records to infer ice volume prior to the LGM. We present a global ice sheet reconstruction for the past 80 000 years, called PaleoMIST 1.0, constructed independently of far-field sea level and δ¹⁸O proxy records. Our reconstruction is compatible with LGM far-field sea-level records without requiring extra ice volume, thus solving the missing ice problem. However, for Marine Isotope Stage 3 (57 000-29 000 years before present) - a pre-LGM period - our reconstruction does not match proxy-based sea level reconstructions, indicating the relationship between marine δ¹⁸O and sea level may be more complex than assumed.