A new Eulerian-Lagrangian coupling on a staggered fluid mesh is proposed to simulate multiscale atomization. This coupling relies on a sharp interface capturing method (ICM) to transport the resolved fluid structures and a Lagrangian tracking algorithm to model the under-resolved Eulerian droplets. The Lagrangian droplets momentum is spread to the source terms of the incompressible fluid momentum equations through a spatial filtering operation, and the flow velocity around the Lagrangian droplets is corrected to account for their local flow disturbance. This allows accurate transport of Lagrangian droplets that are both smaller and larger than the fluid mesh spacing. The implementation of the algorithm for switching from and Eulerian toward Lagrangian framework is discussed, along with criteria validating a transformation. Then the Eulerian-Lagrangian coupling is applied to several test cases from the literature, and is compared to our in-house pure ICM solver on the atomization of a liquid jet. The results show that the Eulerian-Lagrangian coupling improves the physical analysis of the atomization, and achieves more accurate results for poorly resolved droplets.