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We present photometric and spectroscopic observations and analysis of SN 2021bxu (ATLAS21dov), a low-luminosity, fast-evolving Type IIb supernova (SN). SN 2021bxu is unique, showing a large initial decline in brightness followed by a short plateau phase. With $M_r = -15.93 \pm 0.16\, \mathrm{mag}$ during the plateau, it is at the lower end of the luminosity distribution of stripped-envelope supernovae (SE-SNe) and shows a distinct ∼10 d plateau not caused by H- or He-recombination. SN 2021bxu shows line velocities which are at least $\sim 1500\, \mathrm{km\, s^{-1}}$ slower than typical SE-SNe. It is photometrically and spectroscopically similar to Type IIb SNe during the photospheric phases of evolution, with similarities to Ca-rich IIb SNe. We find that the bolometric light curve is best described by a composite model of shock interaction between the ejecta and an envelope of extended material, combined with a typical SN IIb powered by the radioactive decay of 56Ni. The best-fitting parameters for SN 2021bxu include a 56Ni mass of $M_{\mathrm{Ni}} = 0.029^{+0.004}_{-0.005}\, \mathrm{{\rm M}_{\odot }}$, an ejecta mass of $M_{\mathrm{ej}} = 0.61^{+0.06}_{-0.05}\, \mathrm{{\rm M}_{\odot }}$, and an ejecta kinetic energy of $K_{\mathrm{ej}} = 8.8^{+1.1}_{-1.0} \times 10^{49}\, \mathrm{erg}$. From the fits to the properties of the extended material of Ca-rich IIb SNe we find a trend of decreasing envelope radius with increasing envelope mass. SN 2021bxu has MNi on the low end compared to SE-SNe and Ca-rich SNe in the literature, demonstrating that SN 2021bxu-like events are rare explosions in extreme areas of parameter space. The progenitor of SN 2021bxu is likely a low-mass He star with an extended envelope.