We present the first lattice QCD calculation of coupled $\pi \omega$ and $\pi \varphi$ scattering, incorporating coupled $S$ and $D$-wave $\pi \omega$ in $JP=1+$. Finite-volume spectra in three volumes are determined via a variational analysis of matrices of two-point correlation functions, computed using large bases of operators resembling single-meson, two-meson and three-meson structures, with the light-quark mass corresponding to a pion mass of $m\pi \approx 391$ MeV. Utilizing the relationship between the discrete spectrum of finite-volume energies and infinite-volume scattering amplitudes, we find a narrow axial-vector resonance ($JPC=1+-$), the analogue of the $b1$ meson, with mass $mR\approx 1380$ MeV and width $\Gamma R\approx 91$ MeV. The resonance is found to couple dominantly to $S$-wave $\pi \omega$, with a much-suppressed coupling to $D$-wave $\pi \omega$, and a negligible coupling to $\pi \varphi$ consistent with the `OZI rule'. No resonant behavior is observed in $\pi \varphi$, indicating the absence of a putative low-mass $Zs$ analogue of the $Zc$ claimed in $\pi J/\psi$. In order to minimally present the contents of a unitary three-channel scattering matrix, we introduce an $n$-channel generalization of the traditional two-channel Stapp parameterization.