Since the seminal ideas of Berezinskii, Kosterlitz and Thouless, topological excitations have been at the heart of our understanding of a whole novel class of phase transitions. In most cases, those transitions are controlled by a single type of topological objects. There are, however, some situations, still poorly understood, where two dual topological excitations fight to control the phase diagram and the transition. Finding experimental realizations of such cases is thus of considerable interest. We show here that this situation occurs in BaCO₂V₂O₈, a spin-1/2 Ising-like quasi-one-dimensional antiferromagnet, when subjected to a uniform magnetic field transverse to the Ising axis. Using neutron scattering experiments, we measure a drastic modification of the quantum excitations beyond a critical value of the magnetic field. This quantum phase transition is identified, through a comparison with theoretical calculations, to be a transition between two different types of solitonic topological object, which are captured by different components of the dynamical structure factor.