"to synchronize, adjust the phase of so as to synchronize," 1895, from phase (n.) in the physics sense of "particular stage or point in a recurring sequence of movement or changes" (1861). Earlier as a bad spelling of faze. Meaning "to carry out gradually" is from 1949, hence phase in "introduce gradually" (1954), phase out "take out gradually in planned stages" (1954). Related: Phased; phasing.
<a href="https://www.etymonline.com/word/phase-out">Etymology of phase-out by etymonline</a>
Harper, D. (n.d.). Etymology of phase-out. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/phase-out