In this paper, we investigate the propagation of the weak representation property (WRP) to an independently enlarged filtration. More precisely, we consider an F-semimartingale X possessing the WRP with respect to F and an H-semimartingale Y possessing the WRP with respect to H. Assuming that F and H are independent, we show that the G-semimartingale Z= (X, Y) has the WRP with respect to G, where G: = F∨ H. In our setting, X and Y may have simultaneous jump-times. Furthermore, their jumps may charge the same predictable times. This generalizes all available results about the propagation of the WRP to independently enlarged filtrations.