Context. 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, considering the challenges associated with identifying these objects, it is essential to explore complementary selection methods across multiple wavelengths. Aims. 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. The aim is to study the properties of this population, such as their morphological characteristics, and to explore the interplay of SFR and SMBH activity at this epoch. Methods. In order to select the galactic population of our interest, we employed two selection criteria that have frequently been used as separate methods in the literature. In more detail, we selected galaxies that present relatively compact radio morphologies at 1.4 GHz (i.e., an angular size smaller than 10 arcsec) as well as a far-IR spectrum that peaks in flux at λ ≥ 350 μm (i.e., flux_{350 μm} > flux_{250 μm}). For these selection criteria, we used the COSMOS and ECDF-S fields, two of the most extensively observed astronomical fields currently available, which provide high spectral and spatial resolution at a multi-wavelength scale. By accepting only galaxies that satisfied these selection criteria, we derived a sample of eight galaxies that were identified either photometrically or spectroscopically at z > 3 from literature studies and by our team. Results. 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 10³ M_⊙ yr⁻¹ level in massive systems (stellar masses of M_⋆ ∼ 10¹¹ M_⊙) co-hosting an active SMBH. Conclusions. 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. An additional examination of the efficacy of the radio and far-IR selection criteria provides further support for their combined application in selecting co-evolving AGN and star formation activity at high redshift. In this regard, future use of these selection criteria on radio and far-IR/mm observations of statistically larger galaxy samples is of high interest.