Aside from needing a passkey/passphrase every time you open Signal, what would be the solution? If the user can read the unencrypted messages, then so can malware running as the user.

Heck, even if you required some sort of authentication to open the messages, malware could just capture that.

It's the same problem with browser credential stealing, you can grab all the cookies from an authenticated browser session and copy it to a new system.

Really, the biggest issue is that Signal doesn't detect multiple instances running of the same session, but that's also extremely difficult to do without malware being able to work around it.

Not saying there's no solution here, but there is not a simple solution aside from trusting your computer and cancelling sessions if you suspect someone compromised your system (or just not using a desktop app.)