  Next-generation radio experiments such as the Radio Detector of the upgraded
Pierre Auger Observatory and the planned GRAND and BEACON arrays target the
detection of ultra-high-energy particle air showers arriving at low elevation
angles. These inclined cosmic-ray air showers develop higher in the atmosphere
than vertical ones, enhancing magnetic deflections of electrons and positrons
inside the cascade. We evidence two novel features in their radio emission: a
new polarization pattern, consistent with a geo-synchrotron emission model and
a coherence loss of the radio emission, both for showers with zenith angle
$\theta \gtrsim 65^{\circ}$ and strong enough magnetic field amplitude (typical
strength of $B\sim 50\, \rm \mu T$). Our model is compared with both ZHAireS
and CoREAS Monte-Carlo simulations. Our results break the cannonical
description of a radio signal made of Askaryan and transverse current emission
only, and provide guidelines for the detection and reconstruction strategies of
next-generation experiments, including cosmic-ray/neutrino discrimination.
