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Nonreciprocity is most commonly associated with a large difference in the transmitted energy when the locations of the source and receiver are interchanged. This energy bias is accompanied by a difference in the transmitted phase. This work highlights the role of this phase bias in breaking reciprocity in the steady-state vibration transmission characteristics of coupled nonlinear systems in response to external harmonic excitation. It shows that breaking of reciprocity is most commonly accompanied by a simultaneous bias in the transmitted energy and phase. Energy bias alone, without any contribution from phase, can still lead to nonreciprocity, but only at very finely tuned system parameters. A methodology is provided for realizing response regimes of phase-preserving nonreciprocity using two independent symmetry-breaking parameters in the system. The findings highlight the key contribution of phase in nonlinear nonreciprocity.