What comes back?
A return of the phase angle means the moving Moon is once again the same number of degrees ahead of the moving Sun as it was at birth. The angle is directional: 90° is the waxing first-quarter relationship and 270° is the waning last-quarter relationship. It is measured between geocentric apparent ecliptic longitudes. Because the Sun moves roughly a zodiac degree each day, the return does not require either body to occupy its birth longitude.
The cycle repeats about once per synodic month. The calculator searches for the first exact matching angle after the selected UTC date, within a 40-day window. It reports UTC so that a daylight-saving change at a visitor's present location cannot silently change the event. Convert the result to local civil time using the applicable rules for the event location and date.