I am trying to find a way to create a Wii game using SDL2, but I can't find any tutorials on how to do it. How can I use SDL2 to build and privately distribute a Wii game without hacking/modding the Wii?
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You cannot, the Wii does not have a supported environment for developing software, except by licensing with Nintendo and probably renting hardware.
Any private development done is generally on a hacked Wii, there is a SDL port for the Wii if you want to go that route: http://wiibrew.org/wiki/SDL_Wii but it is an incomplete port.