The entire southern sky (Declination, $delta <$ 30 deg) has been observed using the Murchison Widefield Array (MWA), which provides radio imaging of $sim$2-arcmin resolution at low frequencies (72-231 MHz). This is the GaLactic and Extragalactic All-sky MWA (GLEAM) Survey, and we have previously used a combination of visual inspection, cross-checks against the literature, and internal matching to identify the brightest radio-sources ($S_{mathrm{151MHz}} >$ 4 Jy) in the extragalactic catalogue (Galactic latitude, $|b| >$ 10 deg). We refer to these 1,863 sources as the GLEAM 4-Jy (G4Jy) Sample, and use radio images (of $leq$ 45-arcsec resolution), and multi-wavelength information, to assess their morphology and identify the galaxy that is hosting the radio emission (where appropriate). Details of how to access all of the overlays used for this work are available at https://github.com/svw26/G4Jy. Alongside this we conduct further checks against the literature, which we document in this paper for individual sources. Whilst the vast majority of the G4Jy Sample are active galactic nuclei with powerful radio-jets, we highlight that it also contains a nebula, two nearby, star-forming galaxies, a cluster relic, and a cluster halo. There are also three extended sources for which we are unable to infer the mechanism that gives rise to the low-frequency emission. In the G4Jy catalogue we provide mid-infrared identifications for 86% of the sources, and flag the remainder as: having an uncertain identification (129 sources), having a faint/uncharacterised mid-infrared host (126 sources), or it being inappropriate to specify a host (2 sources). For the subset of 129 sources, there is ambiguity concerning candidate host-galaxies, and this includes four sources (B0424$-$728, B0703$-$451, 3C 198, and 3C 403.1) where we question the existing identification.
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