Radiative decay rates are obtained for the first time for allowed (E1) and forbidden (M1, E2) transitions in trebly ionized tungsten (W IV). Our calculations, motivated by strong interest for low-density plasmas and fusion research, illustrate in a convincing way the importance of core-valence correlation effects which substantially increase the lifetimes and, accordingly, decrease the transition probabilities of this heavy ion. Due to the lack of experimental data for W IV, the reliability of the theoretical 'f' values can only be tested by comparison of numerical results obtained with two independent (i.e. multiconfiguration Dirac-Fock and relativistic Hartree-Fock with core-polarization effects (HFR+CPOL)) methods well suited for investigating the atomic structure of heavy ions.