  We generalize previous three-particle finite-volume formalisms to allow for
multiple three-particle channels. For definiteness, we focus on the two-channel
$\eta \pi \pi$ and $K \overline K \pi$ system in isosymmetric QCD, considering
the positive $G$ parity sector of the latter channel, and neglecting the
coupling to modes with four or more particles. The formalism we obtain is thus
appropriate to study the $b_1(1235)$ and $\eta(1295)$ resonances. The
derivation is made in the generic relativistic field theory approach using the
time-ordered perturbation theory method. We study how the resulting
quantization condition reduces to that for a single three-particle channel when
one drops below the upper ($K\overline K \pi$) threshold. We also present
parametrizations of the three-particle K matrices that enter into the
formalism.
