Progress in understanding accreting black holes remains hampered by a lack of sensitive coordinated multiwavelength observations. In particular, the mid-infrared (MIR) regime remains ill-explored except for jet-dominant states. Here, we present comprehensive follow-up of the black hole X-ray binary GX 339─4 during a bright disc-dominated state in its 2023/24 outburst as part of a multiwavelength campaign coordinated around James Webb Space Telescope (JWST)/MIRI. The X-ray properties are fairly typical of soft accretion states with no significant X-ray variability, though with a weak high-energy Comptonized power-law tail. The source is significantly detected, and variable, across 5─10 $\mu$m, at a faint mean flux level. This requires any MIR compact jet contribution to be suppressed by $\gtrsim$ 300 relative to previous hard-state detections. The MIRI spectrum can be described as a simple power-law with slope $\alpha$ = +0.39 $\pm$ 0.07 ($F_\nu$ $\propto$ $\nu ^\alpha$), but matches neither the radio/sub-mm nor the optical broad-band slopes. Synchrotron radiation from the same medium responsible for high-energy Comptonization can self-consistently account for the observed MIRI spectral-timing behaviour, offering new constraints on the physical conditions in the soft-state accretion disc atmosphere/corona. Alternative explanations, including a circumbinary disc, emission from a warm wind, or transient compact jet activity fail to cleanly explain either the spectral properties or the variability. Multiwavelength timing cross-correlations show a puzzlingly long MIR lag relative to the optical, though at limited significance. We compile archival MIR and X-ray luminosities of transient black hole systems, including previously unreported detections of GX 339─4. These trace the evolution of the MIR-to-X-ray flux ratio with accretion state, and also reveal high MIR luminosities for GX 339─4 across all states.