  The origin of the radio synchrotron background (RSB) is currently unknown.
Its understanding might have profound implications in fundamental physics or
might reveal a new class of radio emitters. In this work, we consider the
scenario in which the RSB is due to extragalactic radio sources and measure the
angular cross-correlation of LOFAR images of the diffuse radio sky with matter
tracers at different redshifts, provided by galaxy catalogs and CMB lensing. We
compare these measured cross-correlations to those expected for models of RSB
sources. We find that low-redshift populations of discrete sources are excluded
by the data, while higher redshift explanations are compatible with available
observations. We also conclude that at least 20\% of the RSB surface brightness
level must originate from populations tracing the large-scale distribution of
matter in the universe, indicating that at least this fraction of the RSB is of
extragalactic origin. Future measurements of the correlation between the RSB
and tracers of high-redshift sources will be crucial to constraining the source
population of the RSB.
