Recently the global variation of the Planck mass in the general relativistic Einstein-Hilbert action was proposed as a self-tuning mechanism of the cosmological constant preventing vacuum energy from freely gravitating. We show that this global mechanism emerges for generic local scalar-tensor theories with additional coupling of the scalar field to the field strength of a three-form gauge field that turns the scalar field constant on the domain of the action. Evaluation of the resulting integral constraint equation over the observable Universe yields a self-consistent framework with general relativistic field equations and arbitrary radiatively stable residual cosmological constant. We argue that the expectation value for this residual is in good agreement with the magnitude of the observed cosmic acceleration.