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Amorphous interfacial complexions are particularly resistant to radiation damage and have been primarily studied in alloys with good glass-forming ability, yet recent reports suggest that these features can form even in immiscible alloys such as Cu-Ta under irradiation. In this study, the mechanisms of damage production and annihilation due to primary knock-on atom collisions are investigated for amorphous interphase and grain boundaries in a Cu-Ta alloy using atomistic simulations. Amorphous complexions, in particular amorphous interphase complexions that separate Cu and Ta grains, result in less residual defect damage than their ordered counterparts. Stemming from the nanophase chemical separation in this alloy, the amorphous complexions exhibit a highly heterogeneous distribution of atomic excess volume, as compared to a good glass former like Cu-Zr. Complexion thickness, a tunable structural descriptor, plays a vital role in damage resistance. Thicker interfacial films are more damage-tolerant because they alter the defect production rate due to differences in intrinsic displacement threshold energies during the collision cascade. Overall, the findings of this work highlight the importance of interfacial engineering in enhancing the properties of materials operating in radiation-prone environments and the promise of amorphous complexions as particularly radiation damage-tolerant microstructural features.