Stellar streams are sensitive to perturbations from, for example, giant molecular clouds, bars and spiral arms, infalling dwarf galaxies, or globular clusters which can imprint gaps, clumps, spurs, and asymmetries in tails. In addition to these effects, the impact of a major galactic merger on a population of stellar streams remains to be explored. Here, we focus on the emergence and longevity of asymmetries between the leading and trailing tails of streams caused by such interactions. We ran collisionless N-body simulations of a Milky-Way-like galaxy hosting 36 globular cluster streams and merging with a perturber galaxy. We propose a new asymmetry metric to quantify the structural differences between both tails from their respective cumulative density profiles. We find that the over- and under-densities along streams induced by the merger depend on the orbital characteristics of their progenitors. The non-simultaneity of this effect from stream to stream implies that global asymmetry signatures are less prominent than in individual cases. These population-averaged imprints remain detectable over only 2 Gyr, but asymmetric signatures can persist over much longer periods for individual streams with wide orbits that have been perturbed prior to coalescence. We thus caution that the interpretation of streams' morphology in the context of dark matter mapping is strongly subject to degeneracies and should be performed considering the merger history of the host.