Recent studies suggest that coalescing neutron stars are subject to a fluid instability involving the nonlinear coupling of the tide to p modes and g modes. Its influence on the inspiral dynamics and thus the gravitational wave signal is, however, uncertain because we do not know precisely how the instability saturates. Here we construct a simple, physically motivated model of the saturation that allows us to explore the instability’s impact as a function of the model parameters. We find that for plausible assumptions about the saturation, current gravitational wave detectors might miss >70% of events if only point particle waveforms are used. Parameters such as the chirp mass, component masses, and luminosity distance might also be significantly biased. On the other hand, we find that relatively simple modifications to the point particle waveform can alleviate these problems and enhance the science that emerges from the detection of binary neutron stars.United States. National Aeronautics and Space Administration (ATP Grant NNX14AB40G)National Science Foundation (U.S.)Laser Interferometer Gravitational Wave Observatory