We present a detailed study of the phase properties of rational breather waves observed in the hydrodynamic and optical domains, namely the Peregrine soliton and related second-order solution. At the point of maximum compression, our experimental results recorded in a wave tank or using an optical fiber platform reveal a characteristic phase shift that is multiple of $pi$ between the central part of the pulse and the continuous background, in agreement with analytical and numerical predictions. We also stress the existence of a large longitudinal phase shift across the point of maximum compression.
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