We demonstrate that surface waves propagating at a small angle $2 \theta$ to each other generate large-scale vertical vorticity owing to hydrodynamic nonlinearity in a viscous fluid. The horizontal geometric structure of the induced flow coincides with the structure of the Stokes drift in an ideal fluid, but its amplitude is parametrically larger, it penetrates parametrically deeper into the fluid volume, and its time of evolution is parametrically slower. In an unbounded fluid, the parameter for increasing the amplitude and penetration depth is equal to $1/\sin \theta$, and evolution slows down $1/\sin^2 \theta$ times. We study how the finite depth of the fluid and a thin insoluble liquid film that possibly covers the fluid surface due to contamination affect the generation of large-scale vorticity, and discuss the physical consequences of this phenomenon in the context of wave-driven turbulence.