  The early Universe, spanning 400,000 to 400 million years after the Big Bang
($z\approx1100-11$), has been left largely unexplored as the light from
luminous objects is too faint to be observed directly. While new experiments
are pushing the redshift limit of direct observations, measurements in the
low-frequency radio band promise to probe early star and black hole formation
via observations of the hydrogen 21-cm line. In this work we explore synergies
between 21-cm data from the HERA and SARAS 3 experiments and observations of
the unresolved radio and X-ray backgrounds using multi-wavelength Bayesian
analysis. We use the combined data set to constrain properties of Population II
and Population III stars as well as early X-ray and radio sources. The joint
fit reveals a 68 percentile disfavouring of Population III star formation
efficiencies $\gtrsim5.7\%$. We also show how the 21-cm and the X-ray
background data synergistically constrain opposite ends of the X-ray efficiency
prior distribution to produce a peak in the 1D posterior of the X-ray
luminosity per star formation rate. We find (at 68\% confidence) that early
galaxies were likely 0.3 to 318 times as X-ray efficient as present-day
starburst galaxies. We also show that the functional posteriors from our joint
fit rule out global 21-cm signals deeper than $\lesssim-203\ \mathrm{mK}$ and
power spectrum amplitudes at $k=0.34\ h\mathrm{Mpc^{-1}}$ greater than
$\Delta_{21}^2 \gtrsim 946\ \mathrm{mK}^2$ with $3\sigma$ confidence.
