Motivated by the recent experiment by Schuster et al (Schuster R, Buks E, Heiblum M, Mahalu D, Umansky V and Shtrikman H 1997 Nature 385 417), we study a four-terminal phase-coherent system, each arm having a quantum dot embedded in it (dot 1 for studying and dot 0 for reference). Using the nonequilibrium-Green-function method, the open-circuit collector voltage is derived. We find the following features. (1) The phase behaviours are similar for all of the resonance peaks. (2) In a single resonance peak, the phase increases by on a scale of about the half-peak-width . (3) An abrupt phase drop, by , occurs near the point half way between two consecutive peaks. These results agree well with experiment. We attribute the characteristic (3) to the off-diagonal linewidth of dot 1, which is a single-electron effect. In addition, the crossover of the phase behaviour in going from the four-terminal system to a two-terminal system is studied. Finally, another manifestation of this off-diagonal linewidth is also discussed.