Context :

Aims : Short phases of coeval powerful starburst and AGN activity during the lifetimes of the most massive galaxies are predicted by various models of galaxy formation and evolution .
In spite of their recurrence and high luminosity , such events are rarely observed .
Finding such systems , understanding their nature , and constraining their number density can provide key constraints to galaxy evolutionary models and insights into the interplay between starburst and AGN activities .

Methods : We report the discovery of two sources at z =3.867 and z =3.427 that exhibit both powerful starburst and AGN activities .
They benefit from multi-wavelength data from radio to X rays from the CFHTLS-D1/SWIRE/XMDS surveys .
Follow-up optical and near-infrared spectroscopy , and millimeter IRAM/MAMBO observations are also available .
We performed a multi-wavelength analysis of their spectral energy distributions with the aim of understanding the origin of their emission and constraining their luminosities .
A comparison with other composite systems at similar redshifts from the literature is also presented .

Results : The AGN and starburst bolometric luminosities are \sim 10 ^ { 13 } L _ { \odot } .
The AGN emission dominates at X ray , optical , mid-infrared wavelengths , and probably also in the radio .
The starburst emission dominates in the far-infrared .
The estimated star formation rates range from 500 to 3000 M _ { \odot } /yr .
The AGN near-infrared and X ray emissions are heavily obscured in both sources with an estimated dust extinction { A _ { \mathrm { V } } } \geq 4 , and Compton-thick gas column densities .
The two sources are the most obscured and most luminous AGNs detected at millimeter wavelengths currently known .

Conclusions : The sources presented in this work are heavily obscured QSOs , but their properties are not fully explained by the standard AGN unification model .
In one source , the ultraviolet and optical spectra suggest the presence of outflowing gas and shocks , and both sources show emission from hot dust , most likely in the vicinity of the nucleus .
Evidence of moderate , AGN-driven radio activity is also found in both sources .
Based on the estimated stellar and black hole masses , the two sources lie on the local M _ { BH } - M _ { bulge } relation .
To remain on this relation as they evolve , their star formation rate has to decrease or stop .
Our results support evolutionary models that invoke radio feedback such as the star formation quenching mechanism , and suggest that such a mechanism might play a major role also in powerful AGNs .