We predict a new type of time-resolved cooperative fluorescence (CF) obtainable when a homonuclear alkali-metal diatom dissociating via a $^{1}\mathrm{\ensuremath{\Sigma}}^{\mathrm{*}}$ state evolves (by nonadiabatic radial coupling) into a superposition of two excited adiabatic states correlated to different doublet levels. CF intensity then combines ``ringing'' oscillations, caused by time variation of the interference phase between the receding emitting atoms, with quantum beats associated with fine-structure splitting. The calculated CF patterns for ${\mathrm{Li}}_{2}$ are sensitive to the amplitudes and relative phase of the two superposed adiabatic states.