ABSTRACT The nature of very faint X-ray transients (VFXTs) – transient X-ray sources that peak at luminosities $L_X\lesssim 10^{36} {\rm \, erg \, s^{-1}}$ – is poorly understood. The faint and often short-lived outbursts make characterizing VFXTs and their multiwavelength counterparts difficult. In 2017 April we initiated the Swift Bulge Survey, a shallow X-ray survey of ∼16 square degrees around the Galactic centre with the Neil Gehrels Swift Observatory. The survey has been designed to detect new and known VFXTs, with follow-up programmes arranged to study their multiwavelength counterparts. Here we detail the optical and near-infrared follow-up of four sources detected in the first year of the Swift Bulge Survey. The known neutron star binary IGR J17445-2747 has a K4III donor, indicating a potential symbiotic X-ray binary nature and the first such source to show X-ray bursts. We also find one nearby M-dwarf (1SXPS J174215.0-291453) and one system without a clear near-IR counterpart (Swift J175233.9-290952). Finally, 3XMM J174417.2-293944 has a subgiant donor, an 8.7 d orbital period, and a likely white dwarf accretor; we argue that this is the first detection of a white dwarf accreting from a gravitationally focused wind. A key finding of our follow-up campaign is that binaries containing (sub)giant stars may make a substantial contribution to the VFXT population.