The need for greater energy efficiency has garnered increasing support for the use of fuel-cell technology, a prime example being the solid-oxide fuel cell(1,2). A crucial requirement for such devices is a good ionic (O2- or H+) conductor as the electrolyte(3,4). Traditionally, fluorite- and perovskite-type oxides have been targeted(3-6), although there is growing interest in alternative structure types for intermediate-temperature (400-700 degrees C) solid-oxide fuel cells. In particular, structures containing tetrahedral moieties, such as La1-xCaxMO4-x/2(M= Ta, Nb, P) (refs 7,8), La1-xBa1+xGaO4-x/2 (refs 9,10) and La9.33+xSi6O26+3x/2 (ref. 11), have been attracting considerable attention recently. However, an atomic-scale understanding of the conduction mechanisms in these systems is still lacking; such mechanistic detail is important for developing strategies for optimizing the conductivity, as well as identifying next-generation materials. In this context, we report a combined experimental and computational modelling study of the La1-xBa1+xGaO4-x/2 system, which exhibits both proton and oxide-ion conduction(9,10). Here we show that oxide-ion conduction proceeds via a cooperative 'cog-wheel'-type process involving the breaking and re-forming of Ga2O7 units, whereas the rate-limiting step for proton conduction is intra-tetrahedron proton transfer. Both mechanisms are unusual for ceramic oxide materials, and similar cooperative processes may be important in related systems containing tetrahedral moieties.