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We present the first parton-level study of anomalous effects in triboson production in both fully and semi-leptonic channels in proton-proton collisions at 13TeV at the Large Hadron Collider (LHC). The sensitivity to anomalies induced by a minimal set of bosonic dimension-6 operators from the Warsaw basis is evaluated with specific analyses for each final state. A likelihood-based strategy is employed to assess the most sensitive kinematic observables per channel, where the contribution of Effective Field Theory operators is parameterized at either the linear or quadratic level. The impact of the mutual interference terms of pairs of operators on the sensitivity is also examined. This benchmark study explores the complementarity and overlap in sensitivity between different triboson measurements and paves the way for future analyses at the LHC experiments. The statistical combination of the considered final states allows setting stringent bounds on five bosonic Wilson coefficients.