Can I "trick" software into thinking a remote server is actually on my LAN?

The specific application I have in mind is an annoying game that only supports LAN play through autodetection of other games on your local network, and it won't let you enter an IP address.

So, what I want to do is make it so that myself and two other friends in different locations appear to be on the same local network, so that the game is "fooled" into thinking it's just using local LAN to connect to my friends when it's actually going across the Internet.

I'm having trouble finding details on exactly how to do this, because most people want to do the exact opposite of what I'm trying. They want to make a local machine appear to be part of the public Internet. I'm trying to make a remote machine from outside the network appear to be part of the local network, and there doesn't seem to be as much information on doing that.

Solution 1:

You can use virtual private tunnels (VPNs) for that. VPNs allow you to create a private network tunnel over the Internet with other users, connecting you in a secure local network.
There are applications like Evolve and Hamachi that are designed specifically with that idea in mind - to connect users over the Internet in a private (local) network.

After installing a VPN application, connect with your friends in one server/group, and then, whenever you host a local game, all your friends in that same server/group will be able to see and join your game.

For more information, you can read the following articles:
What is a VPN?
How to play old LAN games over the Internet - despite it being for "old" games, it is a perfect example that can be used for any such game.