Are you married to AnyEvent yet already? Because although I fully agree with doing it the right way with an event loop, I really like Mojo::IOLoop. I was able to take my Mojo::IOLoop TCP server and client example and get TLS working simply by adding the tls=>1 option to the server and tls=>1, tls_options=>{ SSL_verify_mode=>0 } to the client (disabling certificate verification is obviously just for testing). It also seems to work with the example client code you showed here. Mojolicious is also actively maintained, while AnyEvent, despite still being fairly popular as far as I can tell, hasn't had a release in almost four years.

Edit: grammar improvement