  Building on recent NMR experiments [Phys. Rev. Lett. 118, 067203], we
theoretically investigate the high magnetic field regime of the disordered
quasi-one-dimensional $S=1$ antiferromagnetic material
Ni(Cl$_{1-x}$Br$_x$)$_2$-4SC(NH$_2$)$_2$. The interplay between disorder,
chemically controlled by Br-doping, interactions, and the external magnetic
field, leads to a very rich phase diagram. Beyond the well-known
antiferromagnetically ordered regime, analog of a Bose condensate of magnons,
which disappears when $H\ge 12.3$ T, we unveil a resurgence of phase coherence
at higher field $H\sim 13.6$ T, induced by the doping. Interchain couplings
stabilize finite temperature long-range order whose extension in the field -
temperature space is governed by the concentration of impurities $x$. Such a
"mini-condensation" contrasts with previously reported Bose-glass physics in
the same regime, and should be accessible to experiments.
