I have upgraded my SSH-Keys from RSA to Ed25519.
Now if I try to interact with my remote repositories using Sourcetree, the key manager opens up and I unlock the private key with the passphrase. Everything is fine until this point.
Nevertheless Sourcetree can't read from the remote repository anymore.
If I do the same via the shell or any other git interface that supports SSH, everything works fine. So Sourcetree must be the source of this issue.