The performance of a quantum information processor depends on the precise control of phases introduced into the system during quantum gate operations. As the number of operations increases with the complexity of a computation, the phases of the gates at different locations and different times must be controlled, which can be challenging for optically driven operations. We circumvent this issue by demonstrating an entangling gate between two trapped atomic ions that is insensitive to the optical phases of the driving fields while using a common master reference clock for all coherent qubit operations. Such techniques may be crucial for scaling to large quantum information processors in many physical platforms.