Share TLS connections with different databases.

I have a scenario where a database server need to maintain a huge
number of small databases. So to keep TLS clients connected to this
server is a pain because PG requires to have a unique connection for
each database.

For databases deployed in cloud (like AWS) it is even worse, because
we cannot proxy the connections from the database server.

So in order to keep the number of connections acceptable for my env I
usually use a pgbouncer like solution to manage the connections and
reduce the number of connections to the server.

It would be great if PG protocol support reusabilty of TLS connections
and enable clients to switch the database for an established
connection.

I'd like to hear about your experience. Have you seen such scenarios?
How are you dealing with it?
--
Carlo Pires