  Beam-helicity asymmetries have been measured at the MAMI accelerator in Mainz
for the photoproduction of mixed-charge pion pairs in the reactions
$\boldsymbol{\gamma}p\rightarrow n\pi^0\pi^+$ off free protons and
$\boldsymbol{\gamma}d\rightarrow (p)p\pi^0\pi^-$ and
$\boldsymbol{\gamma}d\rightarrow (n)n\pi^0\pi^+$ off quasi-free nucleons bound
in the deuteron for incident photon energies up to 1.4 GeV. Circularly
polarized photons were produced from bremsstrahlung of longitudinally polarized
electrons and tagged with the Glasgow-Mainz magnetic spectrometer. The charged
pions, recoil protons, recoil neutrons, and decay photons from $\pi^0$ mesons
were detected in the 4$\pi$ electromagnetic calorimeter composed of the Crystal
Ball and TAPS detectors. Using a complete kinematic reconstruction of the final
state, excellent agreement was found between the results for free and
quasi-free protons, suggesting that the quasi-free neutron results are also a
close approximation of the free-neutron asymmetries. A comparison of the
results to the predictions of the Two-Pion-MAID reaction model shows that the
reaction mechanisms are still not well understood, in particular at low
incident photon energies in the second nucleon-resonance region.
