The interplay between Brownian colloidal particles and their suspending fluid is well understood since Einstein's seminal work of 1905: the fluid consists of atoms whose thermal motion gives rise to the Brownian motion of the colloids, while the colloids increase the viscosity of the suspension under shear. An alternative route to the viscosity, by exploring the thermal stress fluctuations in a quiescent fluid in the Green-Kubo formalism, however, reveals a marked inconsistency with the viscosity under shear. We show that an additional stress term, accounting for Brownian fluctuating stresslets and coupled to the Brownian forces by a generalized fluctuation-dissipation theorem, is required for the description of the stress and viscosity of a colloidal suspension. Whereas previous applications of the Green-Kubo method to colloidal systems were limited to the deterministic "thermodynamic" part of the stress, using other means to determine the remainder of the viscosity, the whole viscosity is now within the reach of equilibrium studies.