The Moon’s tidal potential is slightly asymmetric, giving rise to so-called third-degree ocean tides, which are small and never before observed on a global scale. High-precision satellite altimeters have collected sea level records for almost three decades, providing a massive database from which tiny, time-coherent signals can be extracted. Here, four third-degree tides are mapped: one diurnal, two semidiurnal, and one terdiurnal. Aside from practical benefits, such as improved tide prediction for geodesy and oceanography, the new maps reveal unique ways the ocean responds to a precisely known, but hitherto unexplored, force. An unexpected example involves the two semidiurnals, where the smaller lunar force is seen to generate the larger ocean tide, especially in the South Pacific. An explanation leads to new information about an ocean normal mode that spatially correlates with the third-degree astronomical potential. The maps also highlight previously unknown shelf resonances in all three tidal bands.