Kinda figured it out:
The problem is psionics/mind control. In late game ground combat with alien psionics (praetors or psionic caesians), sometimes when the alien uses that ability on my soldiers, it would not be successful or something. When it does, it just doesn't do anything. The alien's turn will not resolve for some reason.
A workaround I do is to use the GC editor to move the psionic alien unit near my soldiers and kill him immediately. As soon as he's dead, the game continues as normal. I don't know why this is happening and I don't know how to fix it.