  Skyrmionic materials hold the potential for future information technologies,
such as racetrack memories. Key to that advancement are systems that exhibit
high tunability and scalability, with stored information being easy to read and
write by means of all-electrical techniques. Topological magnetic excitations
such as skyrmions and antiskyrmions, give rise to a characteristic topological
Hall effect. However, the electrical detection of antiskyrmions, in both thin
films and bulk samples has been challenging to date. Here, we apply
magneto-optical microscopy combined with electrical transport to explore the
antiskyrmion phase as it emerges in crystalline mesoscale structures of the
Heusler magnet Mn$_{1.4}$PtSn. We reveal the Hall signature of antiskyrmions in
line with our theoretical model, comprising anomalous and topological
components. We examine its dependence on the vertical device thickness, field
orientation, and temperature. Our atomistic simulations and experimental
anisotropy studies demonstrate the link between antiskyrmions and a complex
magnetism that consists of competing ferromagnetic, antiferromagnetic, and
chiral exchange interactions, not captured by micromagnetic simulations.
