This is why I switched from running daemons in tmux to having them start a slime listener; I can monitor them from any standard process monitor that will log errors and restart (I use daemontools with a run-script that sends me an e-mail to notify me of the restart).

I don't understand the question. Once it's running, it's like any other server, and you monitor it like you would any other server written in any other language.

Speculation here, but based on the context, I think the question is: Some daemon-izer solutions will monitor the daemon and e.g. restart it if it's unresponsive. How do you handle this slash how is this handled in the Common Lisp world?

As a last resort, if Lisp becomes totally unresponsive, you kill the process and restart, same as any other language. But it's pretty rare to lose the REPL, so usually you can fix any unexpected problems through that.