Time dependent large angular spreading and spectral broadening of an intense randomized laser beam propagating in an underdense, well-characterized plasma is measured. The two features are correlated and increase with laser intensity or plasma density. This spatial and temporal incoherence imposed upon the beam via the coupling with the plasma is interpreted, in agreement with recent numerical simulations, as due to the interplay between dynamical filamentation and strongly driven stimulated Brillouin forward scattering.