The net surface water flux (evaporation minus precipitation minus runoff, E-P-R) of the Atlantic Ocean is approximately 0.4 – 0.6 Sv (1 Sv = 10^9 kg s-1) larger than that of the Pacific Ocean, as shown in atmospheric and oceanic reanalyses and by oceanographic estimates. This asymmetry is linked to the asymmetry in sea surface salinity and the existence of the Atlantic Meridional Overturning Circulation. It is shown that the reason for the asymmetry in E-P-R is greater precipitation per unit area over the Pacific south of 30N, while evaporation rates are similar over both basins. It is further argued that the Pacific Ocean is anomalous compared to the Atlantic and Indian Oceans in terms of atmospheric moisture flux convergence and precipitation across the tropics and subtropics. To clarify the mechanism by which water vapour is exported out of the Atlantic basin and imported into the Pacific, we use an air mass trajectory model driven by ERA-Interim reanalysis. Using 12-hourly releases of 14-day back trajectories on the boundaries of ocean drainage basins over the period 2010-2014, we are able to partition the atmospheric moisture fluxes between basins according to their origins (i.e. last contact with the boundary layer). We show that at most a quarter of the E-P-R asymmetry is explained by higher moisture export to the Arctic and Southern basins from the Atlantic than from the Pacific. The main contributions come from differences in the longitudinal atmospheric transport of moisture between the Atlantic, Indian and Pacific basins. In particular, during the Asian summer monsoon the recurvature of the low level flow in the Somali Jet results in a much weaker westward moisture transport from the Indian into the Atlantic basin than across Central America (where it is similar to the zonal average) while there is stronger eastward transport from the Indian to Pacific basins. The net effect is stronger moisture convergence into the Pacific, but weaker into the Atlantic. In contrast to previous thinking, the role of the moisture flux across Central America in the asymmetry, albeit significant, is not dominant.