Measuring the properties of extragalactic magnetic fields through the effect of Faraday rotation provides a means to understand the origin and evolution of cosmic magnetism. Here we use data from the LOFAR Two-Metre Sky Survey (LoTSS) to calculate the Faraday rotation measure (RM) of close pairs of extragalactic radio sources. By considering the RM difference ($Delta$RM) between physical pairs (e.g. double-lobed radio galaxies) and non-physical pairs (i.e. close projected sources on the sky), we statistically isolate the contribution of extragalactic magnetic fields to $Delta$RM along the line of sight between non-physical pairs. From our analysis, we find no significant difference between the $Delta$RM distributions of the physical and non-physical pairs, limiting the excess Faraday rotation contribution to $< 1.9$ rad/m$^2$ ($sim$$95%$ confidence). We use this limit with a simple model of an inhomogeneous universe to place an upper limit of 4 nG on the cosmological co-moving magnetic field strength on Mpc scales. We also compare the RM data with a more realistic suite of cosmological MHD simulations, that explore different magnetogenesis scenarios. Both magnetization of the large scale structure by astrophysical processes such as galactic and AGN outflows, and simple primordial scenarios with seed magnetic field strengths $< 0.5$ nG cannot be rejected by the current data; while stronger primordial fields or models with dynamo amplification in filaments are disfavoured.
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