MIU and CIA are just fancy terms for different model settings or structure settings of the RBC or NK models to introduce money into the models.
In RBC models, MIU with additively separable utility can be super neutrual for money. The crazy stuff is that Under special parameters setting, the FOCs of the MIU model actually produce the Fisher Quantity Identity of Monetary Theory.
CIA setting actually produces non-neutrality of money albeit it is very small. This means that the nominal shock of money supply has real effect on real variables. The FOCs of the model actually relate to the famous Friedman rule in monetary theory by seting the multiplier of CIA constraint equal to zero.