A method is provided involving a fairing for a turbine engine. During this method, a detail is provided. The detail includes a carrier and an exterior layer bonded to the carrier. The carrier is configured from or otherwise includes fiber-reinforced composite material. The exterior layer is configured from or otherwise includes polymer material. The detail is arranged with the fairing. The fairing includes an exterior side and an edge. The detail covers and extends along at least a portion of the exterior side. The detail wraps at least partially around the edge. The carrier and at least a first overhang portion of the exterior layer are bonded to the fairing.