ABSTRACT We use the latest multiredshift (z = 6.5−8.7) upper limits on the 21-cm signal from the Murchison Widefield Array (MWA) to explore astrophysical models which are inconsistent with the data. We explore these limits in the context of reionization astrophysics by using 21CMMC to connect the disfavoured regions of parameter space to existing observational constraints on reionization such as high-z galaxy ultraviolet (UV) luminosity functions, the background UV photoionization rate, the intergalactic medium (IGM) neutral fraction, the electron scattering optical depth and the soft-band X-ray emissivity. We find the vast majority of disfavoured models to already be inconsistent with existing observational constraints. These can be broadly classified into two types of models: (i) ‘cold’ reionization and (ii) pure matter density fluctuations in a cold, neutral IGM (i.e. no reionization). Interestingly, a small subsample of models inconsistent with the MWA is consistent with the aforementioned constraints (excluding the X-ray emissivity). This implies that the current MWA limits are already providing unique information to disfavour models of reionization, albeit extremely weakly. We also provide the first limits on the soft-band X-ray emissivity from galaxies at high redshifts, finding 1σ lower limits of ϵX, 0.5−2 keV ≳ 1034.5 erg s−1 Mpc−3. Finally, we recover 95 per cent disfavoured limits on the IGM spin temperature of $\bar{T}_{\rm S}\lesssim$ 1.3, 1.4, 1.5, 1.8, 2.1, and 2.4 K at z = 6.5, 6.8, 7.1, 7.8, 8.2, and 8.7. With this, we infer the IGM must have undergone, at the very least, a small amount of X-ray heating. Note, the limits on ϵX, 0.5–2 keV and $\bar{T}_{\rm S}$ are conditional on the IGM neutral fraction.