  We obtain analytical expressions for an effective potential of interaction
between two- and three-dimensional (2D and 3D) solitons (including the case of
2D vortex solitons) belonging to two different modes which are incoherently
coupled by cross-phase modulation. The derivation is based on calculation of
the interaction term in the full Hamiltonian of the system. An essential
peculiarity is that, in the 3D case, as well as in the case of 2D solitons with
unequal masses, the main contribution to the interaction potential originates
from a vicinity of one or both solitons, similarly to what was recently found
in the 2D and 3D single-mode systems, while in the case of identical 2D
solitons, the dominating area covers all the space between the solitons. Unlike
the single-mode systems,_stable_ bound states of mutually orbiting solitons are
shown to be possible in the bimodal system.
