Circulation changes have been suggested to play an important role in the sequestration of atmospheric CO2 in the glacial ocean. However, previous studies have resulted in contradictory results regarding the strength of the Atlantic Meridional Overturning Circulation (AMOC) and three-dimensional, quantitative reconstructions of the glacial ocean constrained by multiple proxies remain scarce. Here we simulate the modern and glacial ocean using a coupled physical-biogeochemical, global, three-dimensional model constrained simultaneously by δ13C, radiocarbon, and δ15N to explore the effects of AMOC differences and Southern Ocean iron fertilization on the distributions of these isotopes and ocean carbon storage. We show that δ13C and radiocarbon data sparsely sampled at the locations of existing glacial sediment cores can be used to reconstruct the modern AMOC accurately. Applying this method to the glacial ocean we find that a surprisingly weak (6–9 Sv or about half of today's) and shallow AMOC maximizes carbon storage and best reproduces the sediment isotope data. Increasing the atmospheric soluble iron flux in the model's Southern Ocean intensifies export production, carbon storage, and further improves agreement with δ13C and δ15N reconstructions. Our best fitting simulation is a significant improvement compared with previous studies, and suggests that both circulation and export production changes were necessary to maximize carbon storage in the glacial ocean. •Isotope reconstructions are used to constrain the glacial ocean circulation.•A weak, shallow AMOC and voluminous AABW best reproduce the glacial isotopes.•Higher Southern Ocean export production improve the agreement with reconstructions.•Our best-fitting model to the glacial isotopes maximizes glacial ocean carbon storage.