Spin-polarized currents provide a powerful means of manipulating the magnetization of nanodevices, and give rise to spin transfer torques that can drive magnetic domain walls along nanowires. In ultrathin magnetic wires, domain walls are found to move in the opposite direction to that expected from bulk spin transfer torques, and also at much higher speeds. Here we show that this is due to two intertwined phenomena, both derived from spin-orbit interactions. By measuring the influence of magnetic fields on current-driven domain-wall motion in perpendicularly magnetized Co/Ni/Co trilayers, we find an internal effective magnetic field acting on each domain wall, the direction of which alternates between successive domain walls. This chiral effective field arises from a Dzyaloshinskii-Moriya interaction at the Co/Pt interfaces and, in concert with spin Hall currents, drives the domain walls in lock-step along the nanowire. Elucidating the mechanism for the manipulation of domain walls in ultrathin magnetic films will enable the development of new families of spintronic devices.