  A fundamental question of extra-galactic astronomy that is yet to be fully
understood, concerns the evolution of the star formation rate (SFR) and
supermassive black hole (SMBH) activity with cosmic time, as well as their
interplay and how it impacts galaxy evolution. A primary focus that could shed
more light on these questions is the study of merging systems, comprising
highly star-forming galaxies (SFGs) and active galactic nuclei (AGN) at the
earliest stages of galactic formation. However, it is essential to explore
complementary selection methods across multiple wavelengths. The primary
objective of this study is to conduct a comprehensive analysis of a sample of
high-redshift ($z>3$) far-infrared (far-IR) and radio-emitting galaxies in the
highest possible spatial resolution. In order to select the galactic population
of our interest, we selected galaxies that present relatively compact radio
morphologies at 1.4 GHz as well as a far-IR spectrum that peaks in flux at
$\lambda \geq 350 \, \mu m$. For these selection criteria, we used the COSMOS
and ECDF-S fields, which provide high spectral and spatial resolution at a
multi-wavelength scale. We derived a sample of eight galaxies that were
identified either photometrically or spectroscopically at $z>3$ from literature
studies and by our team. A thorough investigation of available optical,
near-IR, and millimetre (mm) imaging reveals a possible merging scenario in
five out of eight cases in our sample. Additionally, available multi-wavelength
photometry strongly suggests active star formation at the $10^3 \,
M_{\odot}/yr$ level in massive systems co-hosting an active SMBH. Comparison of
these results with previous studies, suggests that our selection method
preferentially identifies galaxies hosting an active SMBH, as well as a strong
SFG component, resulting in high SFR and IR luminosity.
