All credit for this post belongs to one of my colleagues called Imran Mughal, a very talented vSphere engineer!
We recently came across an issue during some site migration work on our PSC’s. The scenario which drove our PSC site migration work was the fact that we noticed all of our vCenter Servers were trying to authenticate users via a single load balanced pair of PSCs regardless of physical location instead of the PSCs in the datacentre local to the vCenter. This was due to us only using a single site name that covered 4 separate datacentre locations. All of our PSCs regardless of which physical location formed part of this single site.
As we already had live service running on out vCenters we decided to re-direct vCenters to another PSCs and rebuild in pairs (as there is currently no way to manually change the site name or create new sites over an existing deployment).
This was done using a combination of these two KB articles: