  The coincident detection of GW170817 in gravitational waves and
electromagnetic radiation spanning the radio to MeV gamma-ray bands provided
the first direct evidence that short gamma-ray bursts (GRBs) can originate from
binary neutron star (BNS) mergers. On the other hand, the properties of short
GRBs in high-energy gamma rays are still poorly constrained, with only $\sim$20
events detected in the GeV band, and none in the TeV band. GRB~160821B is one
of the nearest short GRBs known at $z=0.162$. Recent analyses of the
multiwavelength observational data of its afterglow emission revealed an
optical-infrared kilonova component, characteristic of heavy-element
nucleosynthesis in a BNS merger. Aiming to better clarify the nature of short
GRBs, this burst was automatically followed up with the MAGIC telescopes,
starting from 24 seconds after the burst trigger. Evidence of a gamma-ray
signal is found above $\sim$0.5 TeV at a significance of $\sim3\,\sigma$ during
observations that lasted until 4 hours after the burst. Assuming that the
observed excess events correspond to gamma-ray emission from GRB 160821B, in
conjunction with data at other wavelengths, we investigate its origin in the
framework of GRB afterglow models. The simplest interpretation with one-zone
models of synchrotron-self-Compton emission from the external forward shock has
difficulty accounting for the putative TeV flux. Alternative scenarios are
discussed where the TeV emission can be relatively enhanced. The role of future
GeV-TeV observations of short GRBs in advancing our understanding of BNS
mergers and related topics is briefly addressed.
