Sahara experienced 20 astronomically paced wet periods over past 800kyr. Precession controls African monsoon intensity and NAHP timing. Climate model HadCM3B matches proxy data for NAHPs. Eccentricity modulates NAHP amplitude through ice sheet extent. Glacial ice-albedo cooling suppresses NAHPs during precession minima. High-latitude ice processes determine NAHP strength. NAHPs imply pathways for Out-of-Africa plant and animal dispersal. Models reconcile prior underestimation of NAHP magnitudes. Precession maxima produce strongest Saharan wetness. Vegetation expands across Sahara in NAHPs. No competing interests declared.