Effective field theory provides a way of parameterizing strong-field deviations from General Relativity that might be observable in the gravitational waves emitted in a black hole merger. To perform numerical simulations of mergers in such theories it is necessary that the equations be written in a form that admits a well-posed initial value formulation... We study gravity coupled to a scalar field including the leading (4-derivative) effective field theory corrections. We introduce a new class of "modified harmonic" gauges and gauge-fixed equations of motion, such that, at weak coupling, the equations are strongly hyperbolic and therefore admit a well-posed initial value formulation. (read more)