  The interaction of a magnetic insulator with the helical electronic edge of a
two-dimensional topological insulator has been shown to lead to many
interesting phenomena. One of these is that for a suitable orientation of the
magnetic anisotropy axis, the exchange coupling to an easy-plane magnet has no
effect on DC electrical transport through a helical edge, despite the fact that
it opens a gap in the spectrum of the helical edge [Meng {\em et al.}, Phys.\
Rev.\ B {\bf 90}, 205403 (2014)]. Here, we theoretically consider such a magnet
embedded in an interferometer, consisting of a pair of helical edge states
connected by two tunneling contacts, at which electrons can tunnel between the
two edges. Using a scattering matrix approach, we show that the presence of the
magnet in one of the interferometer arms gives rise to AC currents in response
to an applied DC voltage. On the other hand, the DC Aharonov-Bohm effect is
absent at zero temperature and small DC voltages, and only appears if the
applied voltage or the temperature exceeds the magnet-induced excitation gap.
