A fundamental part of the puzzle of unconventional superconductivity in the Fe-based superconductors is the understanding of the magnetic and nematic instabilities of the parent compounds. The issues of which of these can be considered the leading instability, and whether weak- or strong-coupling approaches are applicable, are both critical and contentious. Here, we revisit the electronic structure of BaFe₂As₂ using angle-resolved photoemission spectroscopy (ARPES). Our high-resolution measurements of samples “detwinned” by the application of a mechanical strain reveal a highly anisotropic 3D Fermi surface in the low-temperature antiferromagnetic phase. By comparison of the observed dispersions with ab initio calculations, we argue that overall it is magnetism, rather than orbital/nematic ordering, which is the dominant effect, reconstructing the electronic structure across the Fe 3d bandwidth. Finally, using a state-of-the-art nano-ARPES system, we reveal how the observed electronic dispersions vary in real space as the beam spot crosses domain boundaries in an unstrained sample, enabling the measurement of ARPES data from within single antiferromagnetic domains, and showing consistence with the effective mono-domain samples obtained by detwinning.