If you were to change the relative phase of some FFT result bins, the place where all the peaks would line up could change, thus representing a time domain shift of some peak. The peaks or transients would be moved to happen earlier or later. Sometimes, an FFT analysis cares about the shape of the time domain waveforms and what time (within the FFT frame) events, peaks, or zero crossings occur. Lossless information thus requires the FFT phase.

Phase in an FFT result also contains information about symmetry: the real or cosine part represents symmetry (even functions around the center), the imaginary component or sine part represent anti-symmetry (odd functions). So any photo or image would get its symmetry hugely distorted without full FFT phase information.

If you were to change the relative phase of some FFT result bins, the place where all the peaks would line up could change, thus representing a time domain shift of some peak. The peaks or transients would be moved to occur earlier or later in the FFT window. Sometimes, an FFT analysis cares about the shape of the time domain waveforms and what time (within the FFT frame) events, peaks, or zero crossings occur. Lossless information thus requires the FFT phase.

Phase in an FFT result also contains information about symmetry: the real or cosine part represents even symmetry (about the center of the FFT aperture), the imaginary component or sine part represent anti-symmetry (an odd function). So any photo or image would get its symmetry hugely distorted without full FFT phase information.