  Nonlinear Compton scattering driven by ultraintense lasers presents a
promising avenue for enhancing the photon energy, brilliance, and setup
compactness of $\gamma$-ray sources. However, a significant challenge lies in
achieving a high polarization degree with commonly generated unpolarized
electrons, thus addressing a longstanding puzzle in the field. Here we
investigate the polarization dynamics of photons emitted by an unpolarized
electron beam interacting with a counter-propagating ultraintense laser pulse
numerically, and propose a novel method to generate highly polarized $\gamma$
rays via nonlinear Compton scattering with the aid of vacuum dichroism effect.
Our simulations reveal that high-brilliance $\gamma$ rays with polarization
beyond 90\% are feasible in a single-shot interaction, rivaling the highest
achieved by any $\gamma$-ray sources to date, based on a developed Monte Carlo
method incorporating polarization-resolved tree processes of nonlinear Compton
scattering and Breit-Wheeler pair production and one-loop vacuum polarization.
This generation method showcases an extraordinary ultra-high polarization
degree and a user-friendly all-optical experimental setup, while harnessing the
high photon energy and brilliance characteristic of nonlinear Compton
scattering sources, thus making it of great potential for experimental
applications.
