In 1981, the idea of a superwind that ends the life of cool giant stars was proposed1. Extreme oxygen-rich giants, OH/IR stars, develop superwinds with the highest mass-loss rates known so far, up to a few 10-4 solar masses (M⊙) per year2-12, informing our understanding of the maximum mass-loss rate achieved during the asymptotic giant branch (AGB) phase. A conundrum arises whereby the observationally determined duration of the superwind phase is too short for these stars to lose enough mass to become white dwarfs2-4,6,8-10. Here we report on the detection of spiral structures around two cornerstone extreme OH/IR stars, OH 26.5 + 0.6 and OH 30.1 - 0.7, thereby identifying them as wide binary systems. Hydrodynamic simulations show that the companion's gravitational attraction creates an equatorial density enhancement mimicking a short, extreme superwind phase, thereby solving the decades-old conundrum. This discovery restricts the maximum mass-loss rate of AGB stars to around the single-scattering radiation pressure limit of a few 10-5 M⊙ yr-1. This has crucial implications for nucleosynthetic yields, planet survival and the wind-driving mechanism.