  We study the polarization properties of the prompt emission of gamma-ray
bursts produced by fragmented fireballs. Such fireballs, known in the
literature under various names, are made by the superposition of many
individual blobs, each of which produces a spike in the light curve. The
differences between pulses are due both to the intrinsic diversity in the
blobs' properties and to their orientation with respect to the line of sight.
We show that the peak flux and the polarization of each pulse are connected
through the orientation of the blob that produces the pulse, while the position
angle fluctuates randomly from one pulse to the next and is constant within an
individual pulse. The most polarized pulses are those with approximately one
tenth the peak flux of the brightest pulse. These conclusions do not depend on
the assumed radiation mechanism nor on the energy and Lorentz factor of the
blobs. We compare the prediction of this model to a simulated set of
observations, showing that a limited sample of GRBs with time-resolved
polarization measurements would provide a crucial test for this model. We
finally show that a hint of the predicted correlation may have been already
observed.
