There are many CB's that perform second party assessments, but it would be a conflict of interest for CB's to be involved with third party certification of companies they audit on behalf of, as well as companies they certify.
Third party certification was developed and envisioned as a means to reduce repetitive, redundant and conflicting second party QMS audits. If it were delivering on its original intent, we would not see the flatlining/drop in certification numbers around some markets.
In order to make the accredited management system certification valuable to Industry at large, somethings need to change. The purpose of the focused audits is to foster better engagement of CB's and the ACTUAL USERS of the services rendered by the CB's and the certificates they issue. Anyone who knows anything about quality understands that the PDCA cycle only works when you have a constructive feedback from relevant stakeholders.
Currently, there is very little engagement and feedback emanating from the registrants customers back to the CB's. I propose, as a means to enhance credibility of the certification sector, that CB's engage, voluntarily or otherwise, much more closely with the users of their certificates.