(Abridged) We study the incidence, as well as the nature, of composite bulges in a sample of 10 face-on barred galaxies to constrain the formation and evolutionary processes of the central regions of disk galaxies. We analyze the morphological, photometric, and kinematic properties of each bulge. Then, by using a case-by-case analysis we identify composite bulges and classify every component into a classical or pseudobulge. In addition, bar-related boxy/peanut (B/P) structures were also identified and characterised. We find only three galaxies hosting a single-component bulge (two pseudobulges and one classical bulge). We find evidence of composite bulges coming in two main types based on their formation: secular-built and merger- and secular-built. We call secular-built to composite bulges made of entirely by structures associated with secular processes such as pseudo bulges, central disks, or B/P bulges. We find four composite bulges of this kind in our sample. On the other hand, merger- and secular-built bulges are those where structures with different formation paths coexist within the same galaxy, i.e., a classical bulge coexisting with a secular-built structure (pseudobulge, central disk, or B/P). Three bulges of this kind were found in the sample. We remark on the importance of detecting kinematic structures such as sigma-drops to identify composite bulges. A large fraction (80%) of galaxies were found to host sigma-drops or sigma-plateaus in our sample revealing their high incidence in barred galaxies. The high frequency of composite bulges in barred galaxies points towards a complex formation and evolutionary scenario. Moreover, the evidence for coexisting merger- and secular-built bulges reinforce this idea. We discuss how the presence of different bulge types, with different formation histories and timescales, can constrain current models of bulge formation.
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