We present a simultaneous study of the phase properties of rational breather waves generated in a water wave tank and in an optical fiber platform, namely, the Peregrine soliton and related second-order solution. Our analysis of experimental wave measurements makes use of standard demodulation and filtering techniques in hydrodynamics and more complex phase retrieval techniques in optics to quantitatively confirm analytical and numerical predictions. We clearly highlight a characteristic phase shift that is a multiple of π between the central pulsed part and the continuous background of rational breathers at their maximum compression. Moreover, we reveal a large longitudinal phase shift across the point of maximum compression.