  We investigate the properties of interacting Q-balls and boson stars that sit
on top of each other in great detail. The model that describes these solutions
is essentially a (gravitating) two-scalar field model where both scalar fields
are complex. We construct interacting Q-balls or boson stars with arbitrarily
small charges but finite mass. We observe that in the interacting case - where
the interaction can be either due to the potential or due to gravity - two
types of solutions exist for equal frequencies: one for which the two scalar
fields are equal, but also one for which the two scalar fields differ. This
constitutes a symmetry breaking in the model. While for Q-balls asymmetric
solutions have always corresponding symmetric solutions and are thus likely
unstable to decay to symmetric solutions with lower energy, there exists a
parameter regime for interacting boson stars, where only asymmetric solutions
exist. We present the domain of existence for two interacting non-rotating
solutions as well as for solutions describing the interaction between rotating
and non-rotating Q-balls and boson stars, respectively.
