High-resolution spectra of Galactic black holes (GBHs) reveal the presence of highly ionized absorbers. In one GBH, accreting close to the Eddington limit for more than a decade, a powerful accretion disc wind is observed to be present in softer X-ray states and it has been suggested that it can carry away enough mass and energy to quench the radio jet. Here we report that these winds, which may have mass outflow rates of the order of the inner accretion rate or higher, are a ubiquitous component of the jet-free soft states of all GBHs. We furthermore demonstrate that these winds have an equatorial geometry with opening angles of few tens of degrees, and so are only observed in sources in which the disc is inclined at a large angle to the line of sight. The decrease in Fe xxv/Fe xxvi line ratio with Compton temperature, observed in the soft state, suggests a link between higher wind ionization and harder spectral shapes. Although the physical interaction between the wind, accretion flow and jet is still not fully understood, the mass flux and power of these winds and their presence ubiquitously during the soft X-ray states suggest they are fundamental components of the accretion phenomenon.