We predict a new state of plasma self-organization, in the presence of magnetic islands, namely turbulence-driven island vortex-flows, which are non-axisymmetric flows. The interaction between an E × B sheared vortex-flow and the drift-wave (DW) turbulence driving it, is derived in the presence of a coherent static magnetic island. The turbulence is driven by a 3D density profile due to quasi-linear island-induced profile flattening, and the DWs thus follow the local electron diamagnetic drift along the island. The metric tensor is introduced, making the analysis easier in island geometry. An extended Charney–Hasegawa–Mima equation describes the DW turbulence—flow interaction, from which a wave-kinetic equation is obtained in island geometry. This yields a turbulence-vortex-flow predator–prey model which predicts a nonlinear threshold for island vortex-flow formation. The vortex-flow threshold decreases with increasing island-width as , which shows that wider islands may more easily drive vortex-flows.