Most bird species are socially monogamous. However, extra‐pair copulations (EPCs), resulting in extra‐pair paternity (EPP), commonly occur. EPCs should allow females to adjust social mate choice and allow males that fail to obtain a nest a chance to avoid missing a breeding season, especially when poor nest supply constrains social mate choice. Procellariiformes (albatrosses and petrels) are socially monogamous seabirds which seldom divorce, even when nest availability constrains social mate choice. In Cory's shearwater Calonectris diomedea, a burrow‐nesting petrel, two studies conducted in the Mediterranean, where competition for nests is weak, detected no EPP. EPP remains to be investigated at localities where competition for nests is much stronger, such as Vila islet, Azores archipelago, Atlantic Ocean. We conducted a genetic (microsatellites) study over two successive years on Vila, involving the breeding pairs of the same 65 nests each year and their single chick. EPPs occurred each year, the overall rate being 11.6%. Coupling genetic analyses to a 7‐year demographic survey provided additional data on pair bonds and competition for nests. Overall, cuckoldry was unrelated to divorce, nest density and inbreeding avoidance, but was more frequent when the social male was small. Nest changes were more costly for males than for females, and some apparently unpaired males attempted to dislodge social males during within‐pair copulations. These results are compatible with the existence of a link between poor nest availability and EPP and confirm that even species considered strongly monogamous can adopt flexible mating strategies.