  Deeply sub-wavelength two-dimensional films may exhibit extraordinarily
strong nonlinear effects. Here we show that 2D films exhibit the remarkable
property of a phase-controllable nonlinearity, i.e., the amplitude of the
nonlinear polarisation wave in the medium can be controlled via the pump beam
phase and determines whether a probe beam will "feel" or not the nonlinearity.
This is in stark contrast to bulk nonlinearites where propagation in the medium
averages out any such phase dependence. We perform a series of experiments in
graphene that highlight some of the consequences of the optical nonlinearity
phase-dependence, {such as} the coherent control of nonlinearly diffracted
beams, single-pump-beam {induced} phase-conjugation and the demonstration of a
nonlinear mirror characterised by negative reflection. The observed phase
sensitivity is not specific to graphene but rather is solely a result of the
dimensionality and is therefore expected in all 2D materials.
