This is the result of a nasty software bug (not my fault though...) due to the upgrade a few weeks ago. For the geeks out there here is the upstream bug report (not created by me): http://bugs.php.net/bug.php?id=52106
While the PHP folks have not been known for prompt bug fixing, this bug is doubly unfortunate because it only occurs for some sites, of which Sibley is one, and consequently there is not a whole lot of interest as of yet.
There are two ways to fix this bug short of PHP fixing it themselves. I can write a workaround or downgrade the software back to the old server's version (both rather hard), or Sibley can provide an alternate way to access their site that does not involve a HTTPS:// connection (my opinion is that HTTPS is unnecessary for a content-based site like Sibley's, but obviously Sibley's tech people do not agree and they may have other considerations).
I'll be keeping a tab on this bug. Unfortunately a solution is unlikely to be forthcoming very soon (as in tomorrow).