EU is economical/societal effort (or war prevention), NATO is military (and doesn't prevent members from fighting, ie Greece vs Turkey). Ever heard of actual EU army? Doesn't exist out there in the world, maybe on paper.
Well, there's Eurocorps for the last few decades, but that's not an army, just a corps-level HQ unit under which component-state forces can be slotted.
OTOH, that's not a particularly poor model for a fairly loose multi-state federation.
On the other other hand, its not actually an EU force, its a force of a particular group of EU states that is made available to both the EU and NATO, and has conducted operations on behalf of each.
