Default instance name different than server name
Sounds like someone thought it would be easy to create a "go-to" SQL Server VM that you can clone. Nobody is bothering to change the standard CO-SQLTEMPLATE SQL Server name. If replication is actually working, I would question whether it is working properly because the SQL Server name should match the VM/Physical server name--minus the instance. What's worse, people are probably connecting to SQL Server using the VM/Physical name and don't know about CO-SQLTEMPLATE.

If you're not going to trash and rebuild, you will probably want to rename SQL Server to match the Windows Server name where it's installed:

exec sp_dropserver 'CO-SQLTEMPLATE'
exec sp_addserver 'MyWIndowsServerName'.


I wouldn't trust that SQL Server setup. If you can, I'd check the Availability Groups folder to see which if the replicated database says myDB (Primary) or myDB (Secondary). Then on the server with myDB (Primary) break and rebuild the AlwaysOn replication and rebuild the replication with the AlwaysOn Wizard. (However, I don't add the listener with the wizard as that part seems buggy to me. I add the listener separate).

But before that, if the servers are as messed up as you say, I'd validate that the Windows cluster is even working properly via Failover Manager. And, validate that DNS actually contains the names for your Windows Servers and your Listener (and possibly a witness)--if you have one.

Caution:
Your servers might even be worse than you imagined with users connecting to both instances of CO-SQLTEMPLATE on two different servers as they both "could" represent two completely different sets of data and are completely unrelated. Because, after all, they used the handy-dandy VM template to make their SQL Server life easier.