I bought the EAW Gold Pack (with Corruption, the expansion) but when I installed, I got the vanilla version (without corruption, no third faction). I tried to force sync with GOG and nothing changed. The manual for Corruption says to play tutorial 8, but there is no tutorial 8 at all.

How do I get the expansion that I bought?
I think I solved it. Under the "more" button in GOG, it shows the expansion's name and launches the gold version.

the gog version defaults to the normal version of the game and as such, you need to run forces of corruption manually. it should however have created an icon. if it did not, just go to its folder and run its executable.