The detection of X-ray emission constitutes a reliable and efficient tool for the selection of active galactic nuclei (AGNs), although it may be biased against the most heavily absorbed AGNs. Simple mid-infrared (IR) broad-band selection criteria identify a large number of luminous and absorbed AGNs, yet again host contamination could lead to non-uniform and incomplete samples. Spectral energy distribution (SED) decomposition is able to decouple the emission from the AGN versus that from star-forming regions, revealing weaker AGN components. We aim to identify the obscured AGN population in the VIMOS Public Extragalactic Redshift Survey in the Canada-France-Hawaii Telescope Legacy Survey W1 field through SED modelling. We construct SEDs for 6860 sources and identify 160 AGNs at a high confidence level using a Bayesian approach. Using optical spectroscopy, we confirm the nature of ~85 per cent of the AGNs. Our AGN sample is highly complete (~92 per cent) compared to mid-IR colour-selected AGNs, including a significant number of galaxy-dominated systems with lower luminosities. In addition to the lack of X-ray emission (80 per cent), the SED fitting results suggest that the majority of the sources are obscured. We use a number of diagnostic criteria in the optical, IR, and X-ray regimes to verify these results. Interestingly, only 35 per cent of the most luminous mid-IR-selected AGNs have X-ray counterparts suggesting strong absorption. Our work emphasizes the importance of using SED decomposition techniques to select a population of type II AGNs, which may remain undetected by either X-ray or IR colour surveys.