We determine elastic and coupled-channel amplitudes for isospin-1 meson-meson scattering in $P$-wave, by calculating correlation functions using lattice QCD with light quark masses such that $m\pi =236$ MeV in a cubic volume of $\sim (4\mathrm{fm})3$. Variational analyses of large matrices of correlation functions computed using operator constructions resembling $\pi \pi$, $KK―$ and $qq¯$, in several moving frames and several lattice irreducible representations, leads to discrete energy spectra from which scattering amplitudes are extracted. In the elastic $\pi \pi$ scattering region we obtain a detailed energy-dependence for the phase-shift, corresponding to a $\rho$ resonance, and we extend the analysis into the coupled-channel $KK―$ region for the first time, finding a small coupling between the channels.