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.