Please do however remember the following line from the link you yourself have cited:
In general, if the behaviour of a system of more than two objects cannot be described by the two-body interactions between all possible pairs, as a first approximation, the deviation is mainly due to a three-body force.
Hence, it can be seen that, initially, when people were thinking of many-body problems, they encountered terms in the mathematical formulation which they later termed as many body forces. Incidentally, they were discovered in strong interactions and were a result of gluon mediation. In the celestial scale hence, you would need to have a similar mediating phenomena/theory to explain any physically valid three body force.
Also related: N-body forces in classical mechanics