Right-dominant handedness is unique and universal in Homo sapiens, suggesting that it is a highly derived trait. Our nearest living relations, chimpanzees, show lateralised hand preference when using tools, but not when otherwise manipulating objects. We report the first contrary data, that is, non-lateralised tool-use, for ant fishing as done in the Mahale Mountains of Tanzania. Unlike nut cracking, termite fishing, and fruit pounding, as seen elsewhere, in which most individuals are either significantly or wholly left- or right-biassed, ant fishers are mostly ambilateral. The clue to this exception lies in arboreality; all other patterns of chimpanzee elementary technology are done on the ground. Arboreal tool use usually requires not only that one hand be used to hold the tool, but also that the other hand gives postural support. When the supporting hand is fatigued, then it must be relieved by the other. Terrestrial tool use entails no such trading off. To test the hypothesis, we compared frequency of hand changing with the incidence of major hand support, and found them to be significantly positively correlated. The evolutionary transition from arboreality to terrestriality may have been a key enabler for the origins of human laterality.