The partition function of a system with pairwise-additive anisotropic dipole-dipole interactions is equal to that of a hypothetical system with many-body isotropic interactions [G. Stell, Phys. Rev. Lett. 32, 286 (1974)]. The effective many-body interactions contain n-body contributions of all orders. Each contribution is known as an expansion in terms of the particle-particle distances r, and the coefficients are temperature dependent. The leading-order two-body term is the familiar −r−6 attraction, and the leading-order three-body term is equivalent to the Axilrod-Teller interaction. In this work, a fluid of particles with the leading-order two-body and three-body interactions is compared to an equivalent dipolar soft-sphere fluid. Molecular simulations are used to determine the conditions under which the effective many-body interactions reproduce the fluid-phase structures of the dipolar system. The effective many-body interaction works well at moderately high temperatures but fails at low temperatures where particle chaining is expected to occur. It is shown that an adjustment of the coefficients of the two-body and three-body terms leads to a good description of the structure of the dipolar fluid even in the chaining regime, due primarily to the ground-state linear configuration of the three-body Axilrod-Teller interaction. The vapor-liquid phase diagrams of systems with different Axilrod-Teller contributions are determined. As the strength of the three-body interaction is increased, the critical temperature and density both decrease and disappear completely above a threshold strength, where chaining eventually suppresses the condensation transition.