The present paper studies the nonhomogeneous incompressible asymmetric fluids in two and three dimensions. The main aim is to obtain the unique global solvability of the system with only bounded nonnegative initial density. More precisely, we construct the global existence of the solution with large data in 2-D and the existence of local in time solution for arbitrary large data and global in time for some smallness conditions in 3-D. In addition, the uniqueness of the solution is proved under quite soft assumptions about its regularity through a Lagrangian approach. These conclusions can be viewed as a generalization of the one established by [ P. Braz e Silva, F. Cruz, M. Loayza and M. Rojas-Medar, J. Differential Equations, 269 (2020), pp. 1319–1348], which does not allow the presence of initial vacuum.