The mechanism responsible for driving the ubiquitous winds of cool giant and supergiant stars remains to be established. To this end, we are constructing semi-empirical models of the extended outer atmospheres (‘chromospheres’) and winds of selected red supergiants. These models are constrained by analyses of the UV line spectra of single stars, and of red supergiants in binaries that eclipse their main-sequence companions: the ζ Aur and VV Cep stars. These detached binaries are well-separated, with no evidence of mass transfer. The C II] 2325 Å line profiles of the binaries are similar to those of comparable single stars, suggesting that the chromospheres remain relatively unperturbed by binarity. However, it is unclear how much binarity disturbs the wind: binary observations suggest a gradual acceleration (β ∼ 3), but line profile analyses of single red supergiants imply a rapid acceleration (β < 1). To date, we have obtained extensive series of HST/GHRS and STIS observations of three eclipsing red supergiant binaries: ζ Aur, HR 2554 and VV Cep. In this paper, we focus on ζ Aur, and present observations and modelling results for this eclipsing binary.