Detached eclipsing binaries are the primary tool used to measure precise masses and radii of stars. In our previous paper estimating the parameters of more than 30 000 detached eclipsing binaries, we identified 766 eclipsing binaries with additional features in their All-Sky Automated Survey for Supernovae (ASAS-SN) and Transiting Exoplanet Survey Satellite light curves. Here, we characterize these 'extra-physics' systems, identifying eclipsing binaries with spotted stars, pulsating components, and candidate triple/quadruple systems. We use the Gaia, Asteroid Terrestrial-impact Last Alert System (ATLAS), Zwicky Transient Facility (ZTF), and ASAS-SN variable star catalogues to search for possible blends. We use MIST (MESA Isochrones and Stellar Tracks) isochrones and evolutionary tracks to identify systems with main-sequence, subgiant, and giant primaries and highlight systems in sparsely populated regions of the colour-magnitude diagram. We find that the orbital period distribution of spotted binaries is divided by evolutionary state and find 68 with X-ray detections. For the candidate triple/quadruples and pulsating systems, we calculate the extra orbital/pulsational period and identify systems with resonances. Finally, we highlight a number of exotic systems, including eclipsing cataclysmic variables, subdwarfs, and binaries with discs.