Originally Posted by
dulciusexasperis
So leave US on US passport(actually required by law of all US citizens); enter Ireland on her Irish Passport (there is no such thing as an EU passport); leave Ireland on her Irish Passport; enter US on her US passport.
Well, yes except there are no routine emigration checks when leaving the US or Ireland, so when leaving Ireland you need to show your US passport to the airline otherwise they will not let you board a flight to the US. Not so much an issue in the other direction for the moment. Of course if US CBP decides to do a random passport check on leaving the US you should show them your US passport.
Another example is where entering on one passport may require a Visa (and paying for one) while entering on another passport may be done on a 'Visa Waiver' (which costs nothing).
Again true except with the current case in point, a US 'visa waiver' does not cost nothing.