A higher-order interaction is irreducible when the vector field of one unit depends jointly on two or more other units, in a way that no sum of pairwise terms reproduces. Inferring such couplings from time series of coupled oscillators faces two separate obstructions. The first is identifiability. We measure it with an order-resolved indicator, the smallest singular value of the three-body block of the design matrix after projection onto the complement of all other blocks. We prove that this indicator vanishes on every phase-locked trajectory, that an intervention restores it for almost every system, and that a small kick restores it only in proportion to the square of its amplitude. With ensembles of independently prepared states, every irreducible coupling of a four-oscillator network is detected at phase noise up to 0.03, whereas single passive trajectories detect about half of them. In mean-field networks of up to 50 units the number of prepared states needed grows in proportion to the number of three-body coefficients per unit, each preparation resolving about seven of them. The second obstruction is attribution. Phase reduction of pairwise Stuart–Landau networks generates effective three-body terms, whose coefficients we derive and recover quantitatively with the same ensemble design. A short-time intervention separates physical from effective three-body terms: the non-additive response to paired kicks grows linearly with the response time for an irreducible coupling, and quadratically when the three-body term is mediated by a hidden node or by eliminated amplitudes.