Observations show that the flow of Rutford Ice Stream (RIS) is strongly modulated by the ocean tides, with the strongest tidal response at the 14.77-day tidal period (Msf). This is striking because this period is absent in the tidal forcing. A number of mechanisms have been proposed to account for this effect, yet previous modelling studies have struggled to match the observed large amplitude and decay length scale. We use a nonlinear 3-D viscoelastic full-Stokes model of ice-stream flow to investigate this open issue. We find that the long period Msf modulation of ice-stream velocity observed in data cannot be reproduced quantitatively without including a coupling between basal sliding and tidally induced subglacial water pressure variations, transmitted through a highly conductive drainage system at low effective pressure. Furthermore, the basal sliding law requires a water pressure exponent that is strongly nonlinear with q = 10 and a nonlinear basal shear exponent of m = 3. Coupled model results show that sub-ice shelf tides result in a ∼12 % increase in mean horizontal velocity of the adjoining ice stream. Observations of tidally induced variations in flow of ice streams provide stronger constraints on basal sliding processes than provided by any other set of measurements.