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Connecting to an external database is no different than a localhost db, you only need to specify where it is. And have the db user configured to be allowed access from the web server (MySQL permissions).
And, you need to be able to route MySQL between the servers (no firewall issues etc.). Now, are you saying that you are routing MySQL across the internet to a home database server? Certainly possible but if it is across the open internet that isn't wise or safe. Always have at least the web server between user and data.
Push it over a VPN or some way that prevents exposing your database to anything but local traffic (at most) and the web server that needs access. Even with server and db on the same box it is bad to allow access from random internet users no matter how tight your user access is set.
I'll repeat:
Now, are you saying that you are routing MySQL across the internet to a home database server?
Don't and don't unless it is protected very well!
