The evolution of past global ice sheets is highly uncertain. One example is the missing ice problem during the Last Glacial Maximum (LGM, 26000-19000 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 delta O-18 proxy records to infer ice volume prior to the LGM. We present a global ice sheet reconstruction for the past 80000 years, called PaleoMIST 1.0, constructed independently of far-field sea level and delta O-18 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 (57000-29000 years before present) - a pre-LGM period - our reconstruction does not match proxy-based sea level reconstructions, indicating the relationship between marine delta O-18 and sea level may be more complex than assumed. The configuration of past ice sheets, and therefore sea level, is highly uncertain. Here, the authors provide a global reconstruction of ice sheets for the past 80,000 years that allows to test proxy based sea level reconstructions and helps to reconcile disagreements with sea level changes inferred from models.