Bosons in one dimension display a phenomenon called quasicondensation where correlations decay in a power-law fashion. We study the fate of this quasicondensed state in an open-system setup where two macroscopic leads, held at different chemical potentials, drive the system far away from equilibrium. For generic leads, we find that a finite bias destroys the quasicondensed state and the system exhibits exponential-decay correlations and subleading power-law corrections that are bias dependent. Near the equilibrium, we found a diverging correlation length and determine the critical properties, including the critical scaling form of the correlations. Also, we contrast our findings against the fate of quasicondensation in thermal equilibrium and discuss the special case of reflectionless coupling where the quasicondensation survives out of equilibrium. This exactly solvable interacting nonequilibrium system has the remarkable property that the near-equilibrium state cannot be obtained within linear response. These results aid in unraveling the enigmatic properties spawned by strong interactions once liberated from equilibrium constraints.