  Many-body forces are sometimes a relevant ingredient in various fields, such
as atomic, nuclear or hadronic physics. Their precise structure is generally
difficult to uncover. So, phenomenological effective forces are often used in
practice. Nevertheless, they are always very heavy to treat numerically. The
envelope theory, also known as the auxiliary field method, is a very efficient
technique to obtain approximate, but reliable, solutions of many-body systems
interacting via one- or two-body forces. It is adapted here to allow the
treatment of a special form of many-body forces. In the most favourable cases,
the approximate eigenvalues are analytical lower or upper bounds. Otherwise,
numerical approximation can always be computed. Two examples of many-body
forces are presented, and the critical coupling constants for generic
attractive many-body potentials are computed. Finally, a semiclassical
interpretation is given for the generic formula of the eigenvalues.
