Quantum theory predicts the existence of genuinely tripartite-entangled states, which cannot be obtained from local operations over any bipartite-entangled states and unlimited shared randomness. Some of us recently proved that this feature is a fundamental signature of quantum theory. The state |GHZ_{3}⟩=(|000⟩+|111⟩)/sqrt[2] gives rise to tripartite quantum correlations that cannot be explained by any causal theory limited to bipartite nonclassical common causes of any kind (generalizing entanglement) assisted with unlimited shared randomness. Hence, any conceivable physical theory that would reproduce quantum predictions will necessarily include genuinely tripartite resources. In this Letter, we verify that such tripartite correlations are experimentally achievable. We derive a new device-independent witness capable of falsifying causal theories wherein nonclassical resources are merely bipartite. Using a high-performance photonic |GHZ_{3}⟩ state with fidelities of 0.9741±0.002, we provide a clear experimental violation of that witness by more than 26.3 standard deviations, under the locality and fair sampling assumption. We generalize our Letter to the |GHZ_{4}⟩ state, obtaining correlations that cannot be explained by any causal theory limited to tripartite nonclassical common causes assisted with unlimited shared randomness.