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Droplet breakup in immiscible polymer blends after a sudden substantial increase in shear rate is studied using small-angle light scattering experiments. During the breakup of the fibrils, secondary streaks are visible in the scattering pattern, which indicate the occurrence of Rayleigh instabilities. A high-resolution camera allows quantitative determination of the evolution of the wavelength of the disturbances during the disintegration process. It is shown that this process is time-controlled rather than strain-controlled as suggested by theory.