We present the first calculation of coupled-channel meson-meson scattering in the isospin $=1$, $G$-parity negative sector, with channels $pi eta$, $Koverline{K}$ and $pi eta$, in a first-principles approach to QCD. From the discrete spectrum of eigenstates in three volumes extracted from lattice QCD correlation functions we determine the energy dependence of the $S$-matrix, and find that the $S$-wave features a prominent cusp-like structure in $pi eta to pi eta$ close to $Koverline{K}$ threshold coupled with a rapid turn on of amplitudes leading to the $Koverline{K}$ final-state. This behavior is traced to an $a_0(980)$-like resonance, strongly coupled to both $pi eta$ and $Koverline{K}$, which is identified with a pole in the complex energy plane, appearing on only a single unphysical Riemann sheet. Consideration of $D$-wave scattering suggests a narrow tensor resonance at higher energy.
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