Savannas cover a significant fraction of the Earth's land surface. In these ecosystems, C3 trees and C4 grasses coexist persistently, but the mechanisms explaining coexistence remain subject to debate. Different quantitative models have been proposed to explain coexistence, but these models make widely contrasting assumptions about which mechanisms are responsible for savanna persistence. Here, we show that no single existing model fully captures all key elements required to explain tree-grass coexistence across savanna rainfall gradients, but many models make important contributions. We show that recent empirical work allows us to combine many existing elements with new ideas to arrive at a synthesis that combines elements of two dominant frameworks: Walter's two-layer model and demographic bottlenecks. We propose that functional rooting separation is necessary for coexistence and is the crux of the coexistence problem. It is both well-supported empirically and necessary for tree persistence, given the comprehensive grass superiority for soil moisture acquisition. We argue that eventual tree dominance through shading is precluded by ecohydrological constraints in dry savannas and by fire and herbivores in wet savannas. Strong asymmetric grass-tree competition for soil moisture limits tree growth, exposing trees to persistent demographic bottlenecks.