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[Display omitted] Many-body forces play a prominent role in structure and dynamics of matter, but their role is not well understood in many cases due to experimental challenges. Here, we demonstrate that a novel experimental system based on rotating electric fields can be utilised to deliver unprecedented degree of control over many-body interactions between colloidal silica particles in water. We further show that we can decompose interparticle interactions explicitly into the leading terms and study their specific effects on phase behaviour. We found that three-body interactions exert critical influence over the phase diagram domain boundaries, including liquid-gas binodal, critical and triple points. Phase transitions are shown to be reversible and fully controlled by the magnitude of external rotating electric field governing the tunable interactions. Our results demonstrate that colloidal systems in rotating electric fields are a unique laboratory to study the role of many-body interactions in physics of phase transitions and in applications, such as self-assembly, offering exciting opportunities for studying generic phenomena inherent to liquids and solids, from atomic to protein and colloidal systems. [ABSTRACT FROM AUTHOR]