I have followed the “official” directions on configuring TLS and also various configuration tweaks others have suggested on IRC and Google search results.
It is not a networking problem. The phone is connected directly to the PBX. I can ping the phone and Nmap shows the relevant port is open to TCP connections.
Therefore it must be a problem with the phone, the asterisk box or both.
The only option left I can think of is to test if there is an issue with the version of Asterisk or Open SSL used. There isn’t much room to move on the phone as it is already running the latest firmware, and there is no configuration option I can find that is significant beyond the SIP server (host.domain), port (5061) and type (TLS). I installed the Asterisk server CA cert and a cert on the phone.
In short I would have to say I am at my wits end. I place this up there with one of the most troublesome problems ever.