This paper analyses numerically the quasi-two-dimensional flow of an incompressible electrically conducting viscous fluid past a localized zone of applied magnetic field, denominated a magnetic obstacle. The applied field is produced by the superposition of two parallel magnetized square surfaces, uniformly polarized in the normal direction, embedded in the insulating walls that contain the flow. The area of these surfaces is only a small fraction of the total flow domain. By considering inertial effects in the analysis under the low magnetic Reynolds number approximation, it is shown that the flow past a magnetic obstacle may develop vortical structures and eventually instabilities similar to those observed in flows interacting with bluff bodies. In the small zone where the oncoming uniform flow encounters the non-negligible magnetic field, the induced electric currents interact with the field, producing a non-uniform Lorentz force that brakes the fluid and creates vorticity. The effect of boundary layers is introduced through a friction term. Due to the localization of the applied magnetic field, this term models either the Hartmann braking within the zone of high magnetic field strength or a Rayleigh friction in zones where the magnetic field is negligible. Finite difference numerical computations have been conducted for Reynolds numbers $\hbox{\it Re}\,{=}\,100$ and 200, and Hartmann numbers in the range $1 \le \hbox{\it Ha} \le 100$ ($\hbox{\it Re}$ and $\hbox{\it Ha}$ are based on the side length of the magnetized square surfaces). Under these conditions, a wake is formed behind the obstacle. It may display two elongated streamwise vortices that remain steady as long as the Hartmann number does not exceed a critical value. Once this value is reached, the wake becomes unstable and a vortex shedding process similar to the one observed in the flow past bluff bodies is established. Similarities and differences with the flow around solid obstacles are discussed.