  We consider a domain wall in the mesoscopic quasi-one-dimensional sample
(wire or stripe) of weakly anisotropic two-sublattice antiferromagnet, and
estimate the probability of tunneling between two domain wall states with
different chirality. Topological effects forbid tunneling for the systems with
half-integer spin S of magnetic atoms which consist of odd number of chains N.
External magnetic field yields an additional contribution to the Berry phase,
resulting in the appearance of two different tunnel splittings in any
experimental setup involving a mixture of odd and even N, and in oscillating
field dependence of the tunneling rate with the period proportional to 1/N.
