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Integrated optical affinity sensors based on Mach-Zehnder interferometers (MZIs) enable sensitive and label-free detection of chemical and biological molecules. However, since MZIs transduce optical phase changes into sinusoidal intensity variations, they exhibit a variable sensitivity that vanishes at the extrema of the transmittance function. Moreover, the direction of phase change cannot be unambiguously determined with these sensors. Here, we present a coherent detection scheme based on integrated optics that enables unambiguous readout of the optical phase with a constant sensitivity. Our approach furthermore cancels the effect of imperfections in the sensor hardware using a blind calibration scheme. We experimentally show completely linear readout of the optical phase, with a fourfold enhancement of average sensitivity compared to conventional detection.