Acoustic levitation in air provides a containerless, gravity-free platform for investigating driven many-particle systems with nonconservative interactions and underdamped dynamics. In prior work the interactions among levitated particles were limited to attractive forces from scattered sound and repulsion from hydrodynamic microstreaming. We report on experiments in which contact cohesion provides a third type of interaction. When particle size and separation are both much smaller than the sound wavelength, this interplay of three interactions results in forces that are attractive over several particle diameters, become repulsive at close approach, and are again attractive at contact. In the presence of sound-induced athermal fluctuations that generate particle collisions, the interplay of these three forces enables the formation of particle chains with anisotropic interactions that depend on chain size and shape due to multibody effects. With the control of the kinetic pathways and the strength of the contact cohesion, different patterns can be assembled, from triangular lattices to labyrinthine patterns of chains to lacelike networks of interconnected rings. These results shed light on the multibody character of acoustic interactions and can be utilized to direct the self-assembly of particles.