Hi @Jörg Mai,
Thank you for posting to Microsoft Community. We are happy to help you.
Based on your description, I understand the issue is the Outlook continually prompts for login. However, I am confused about what you said about “the cloud accesses an on-prem account and vice versa”. Could you kindly explain it to me for understanding and helping you better.
The login prompt issue can be caused by a variety of factors, including incorrect login credentials, outdated software, or a corrupted Outlook profile. Exclude attempts you've already made. You could perform comprehensive troubleshooting such as the following.
- Corrupt Outlook Profile. Corrupt profile may cause Outlook to no longer remember your password. Please try to create a new mail profile in Control Panel and reconfigure your email account in the new profile to fix the issue. Create an Outlook profile - Microsoft Support.
- Kindly run Microsoft Support and Recovery Assistant (SaRA Tool) About the Microsoft Support and Recovery Assistant - Microsoft Support
- Launch Outlook in Safe Mode as following link Open Outlook in safe mode - Microsoft Support. This will disable any add-ins that may be causing the issue.
- Disable the Always prompt for login option in Outlook.
- Incorrect credential cached in Credential Manager. If the cached credentials are incorrect, you will be prompted for password. Try to remove incorrect credentials from the Credentials Manager.
Also, could you kindly share the login prompt screenshot for the further investigation.
If there are anything else you need help, please feel free to reply to me.