Coronal-line forest (CLiF) active galactic nuclei (AGNs) are characterized by strong high-ionization lines, which contrasts with what is found in most AGNs. Here, we carry out a multiwavelength analysis aimed at understanding the physical processes in the narrow-line region (NLR) of these objects, and at discovering whether they are indeed a special class of AGNs. By comparing coronal emission-line ratios we conclude that there are no differences between CLiF and non-CLiF AGNs. We derive physical conditions of the NLR gas and we find electron densities in the range of 3.6 × 102 to 1.7 × 104 cm-3 and temperatures of 3.7 × 103 to 6.3 × 104 K, suggesting that the ionization mechanism is associated primarily with photoionization by the AGN. We suggest an NLR dominated by matter-bounded clouds to explain the high-ionization line spectrum observed. The mass of the central black hole, derived from the stellar velocity dispersion, shows that most of the objects have values in the interval 107-108 M☉. Our results imply that CLiF AGNs are not in a separate category of AGNs. In all optical/near-infrared emission-line properties analysed, they represent an extension to the low/high ends of the distribution within the AGN class.