We measure the degree of coherence of supercontinua generated in tapered fibers by subsequent fs pulses. By means of interference experiments we study its dependence on the input pulse duration and power. We also present numerical simulations that allow us to explain the experimental observations which show a decreasing degree of coherence with increasing input power. We attribute this loss of coherence to phase noise due to the cross-phase modulation by several solitons with randomly varying parameters due to quantum noise.