  Time-resolved linear polarization ($\Pi$) measurements of the prompt
gamma-ray burst emission can reveal its dominant radiation mechanism. A widely
considered mechanism is synchrotron radiation, for which linear polarization
can be used to probe the jet's magnetic-field structure, and in turn its
composition. In axisymmetric jet models the polarization angle (PA) can only
change by $90^\circ$, as $\Pi$ temporarily vanishes. However, some
time-resolved measurements find a continuously changing PA, which requires the
flow to be non-axisymmetric in at least one out of its emissivity, bulk Lorentz
factor or magnetic field. Here we consider synchrotron emission in
non-axisymmetric jets, from an ultrarelativistic thin shell, comprising
multiple radially-expanding mini-jets (MJs) or emissivity patches within the
global jet, that yield a continuously changing PA. We explore a wide variety of
possibilities with emission consisting of a single pulse or multiple
overlapping pulses, presenting time-resolved and integrated polarization from
different magnetic field configurations and jet angular structures. We find
that emission from multiple incoherent MJs/patches reduces the net polarization
due to partial cancellation in the Stokes plane. When these contain a
large-scale ordered field in the plane transverse to the radial direction,
$\Pi$ always starts near maximal and then declines over the single pulse or
shows multiple highly polarized peaks due to multiple pulses. Observing
$\Pi\lesssim40\%$ ($15\%$) integrated over one (several) pulse(s) will instead
favor a shock-produced small-scale field either ordered in the radial direction
or tangled in the plane transverse to it.
