  Since the seminal ideas of Berezinskii, Kosterlitz and Thouless, topological
excitations are at the heart of our understanding of a whole novel class of
phase transitions. In most of the 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 realization of such
cases is thus of considerable interest. We show here that this situation occurs
in BaCo$_2$V$_2$O$_8$, 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 objects, which are captured by different components of
the dynamical structure factor.
