For relativistic Cherenkov devices we investigate the process of high-power microwave (HPM) pulse generation with its phase correlating to the sharp edge of an e-beam current pulse. Our theoretical consideration is referred to quasistationary and superradiative (SR) generation regimes when spontaneous emission of the e-beam edge serves as the seed for the development of further coherent oscillations. Phase correlation of the excited microwave pulses with the characteristics of the current pulse front and/or an initial external electromagnetic pulse provided additional confirmation of particle-in-cell (PIC) simulations. Pulse-to-pulse stability of the radiation phase within several percent of the oscillation period makes it possible to arrange multi-channel schemes producing mutually coherent microwave pulses. In the experiments that have been carried out, the cathodes of independent generators were powered by identical accelerating pulses from strictly synchronized voltage modulators, or by splitting the pulse from a single powerful modulator. For the 2-ns regime with the power of each Ka-band backward wave oscillator (BWO) about 100 MW, we demonstrate quadratic growth of the power density in the interference maximum of the directional diagram. In a short pulse SR regime, with the peak power of 600 MW in a single channel, for a four-channel 2-D array we attained a sixteen-fold radiation intensity gain.