How can I connect to AOE2 over the Internet?

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There is always more than one way to do something and the same applies here. I have found a way that is easy and works great, so I am using it as an example.

First your host machine and you need to install
The tutorial is pretty good at walking you through the options.

You will then need to run AOE and go to the Internet TCP/IP connection in the multiplayer menu. Double click brings up a place to type an IP or host machine name. You can get the host machine name from Hamachi or just ask the server host what their information is. You should be able to connect.

And that's it. Pretty easy and straightforward.