If you only have one map that you want this, having the Parallel event as @Aerosys
indicated above would be cool. Or maybe a couple of maps.
However, if you wanted this effect on many maps and didn't want to have a Parallel event on every map, you could also consider having the coat as an Item. With an Item, when you use it, you can call a Common Event.
So you could set the Common Event up that when you use the Item, you get text that says the Coat is put on and a Switch is enabled. When you use the Item again, the Common Event would have a check to see if the Switch is on. If so, then the Common Event would ask if you want to remove the Coat. If so, you get text that the coat is removed and the Switch is turned off.
That way you only need the item, a Common Event, and a Switch, but no parallel event.
Of course you wouldn't see the coat as Equipped this way.