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Galactic outflows driven by star formation and active galactic nuclei blow bubbles into their local environments, causing galactic magnetic fields to be carried into intergalactic space. We explore the redshift-dependent effect of these magnetized bubbles on the Faraday Rotation Measure (RM) of extragalactic radio sources. Using the IllustrisTNG cosmological simulations, we separate the contribution from magnetic bubbles from that of the volume-filling magnetic component expected to be due to the seed field originating in the Early Universe. We use this separation to extract the redshift dependence of each component and to compare TNG model predictions with observation measurements of the NRAO VLA Sky Survey (NVSS). We find that magnetized bubbles provide a sizeable contribution to the extragalactic RM, with redshift-independent $\langle |{\rm RM}| \rangle \simeq 13$ rad/m$^2$ for sources at redshifts $z\ge 2$. This is close to the mean residual RM of $16$ rad/m$^2$ found from NVSS data in this redshift range. Using the IllustrisTNG simulations, we also evaluate a simple model for the contribution to residual RM from individual host galaxies and show that this contribution is negligible at high-redshift. While the contribution from magnetic bubbles in the IllustrisTNG model is currently compatible with observational measurements of residual RM, the next-generation RM sky surveys, which will be free from the wrapping uncertainty, have larger statistics and better sensitivity should be able to observe predicted flat contribution from magnetic bubbles at large redshifts. This should allow to experimentally probe magnetic bubbles and check models of galaxy feedback in cosmological simulations.