We consider a hydrodynamic multi-phase field problem to model the interaction of deformable objects. The numerical approach considers one phase field variable for each object and allows for an independent adaptive mesh refinement for each variable. Using the special structure of various terms allows interpolating the solution on one mesh onto another without loss of information. Together with a general multi-mesh concept for the other terms speedup by a factor of two can be demonstrated which improves with the number of interacting objects. The general concept is demonstrated on an example describing the interaction of red blood cells in an idealized vessel.