I am facing an issue connecting to my GCP Cloud SQL MySQL database using a Private IP address from MySQL Workbench.
I ran into a similar issue connecting to Cloud SQL over Private IP. A couple things helped:
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Make sure your local machine is connected to the same VPC network using Cloud VPN or Cloud Interconnect (Private IP won’t work directly over the public internet).
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Confirm that your VPC firewall rules allow MySQL traffic on port 3306.
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Double-check the authorized networks and that the instance actually has a Private IP assigned.
Once the network path was in place, MySQL Workbench connected without any problems.