Turbulence is commonly regarded as the ultimate state of disorder, in which coherent structures do not persist in time averages. Here we show that fully developed electrohydrodynamic turbulence in nematic liquid crystals instead supports a robust, system-spanning mean flow. Using high-speed imaging and particle tracking, we uncover a persistent large-scale vortex that emerges spontaneously from the dynamic scattering mode-2 (DSM2), a turbulent state studied for over five decades. This mean flow survives temporal averaging and reflects genuine material transport rather than apparent motion of orientational patterns. Three-dimensional nematohydrodynamic simulations reproduce the observed flow and reveal a Poiseuille-like velocity profile with vanishing interfacial shear stress, while two-dimensional modeling captures the coarsening dynamics leading to vortex selection. The emergence of a zero-wavenumber flow directly from fully developed turbulence contrasts sharply with previously studied active and classical turbulent systems, where large-scale organization typically arises at finite wavelengths or requires external forcing.