Credentials for each connection need to be entered separately into Tableau Bridge in order for the refresh to succeed. See Step 3 "Embed database credentials into the connection information" in Tableau Help: Publishers: Use Tableau Bridge to Keep Tableau Cloud Data Fresh.
Datasources with cross-database joins that have connections which are not configured to be refreshed by Tableau Bridge Client may fail with the error NO_POOLED_AGENTS_ASSIGNED. In this scenario, that error occurs because all connections are must use Tableau Bridge, and one or more of the datasources does not match a domain in the Private Network Allowlist. See more about this error in Tableau Help: Configure the Bridge Client Pooling - Troubleshooting
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