A quick (OK, rather longish) situation to explain why I disagree with the FAQ.
A Fighter with Cleave attacks a wizard with mirror image cast upon him.
The fighter is not a total idiot - he has adventured before and knows wizards casts spells.
He sees six seperate images of the exact same person - who is dressed like a wizard and holding a but of bat turd like wizards do right before that spell that almost keeps killing the fighter and his allies every time they fight a wizard is cast. He kinda figures that this wizard looking antagonist is (shock) a wizard, and the multiple images of him are obviously a spell that was cast. That, or his party is being attacked by cloned wizards that all move and act in unison. He decides the first level spell is more likely the case, and doubts the party is being attacked by an entire litter of synchronized identical sibling wizards.
So, knowing that he is fighting a wizard and that some sort of illusion must have been cast if he us currently viewing six of them, he swings at the closest one and connects. Well, he would have connected... but it's an illusion, and not really there. Not feeling his blade strike anything, he, stops the swing because he knows he has not actually connected to anything tangible. he would not continue a full follow through with this attack, there is no actual reason to do so. Unless the Fighter is an idiot, or mirror image somehow makes duplicates of the spel's caster that are NOT identical to him, so that someone seeing mirror image might not be able to figure out what's going on - mirror image being one of the speels I am personally of the opinion that only a moron thinks you need a spellcraft check to figure out what happened. Um, he cast a spell, not there are more of him, but they still move when he does. Duh? Anyway....
So, not following through on a swing that never connected there is no chance to cleave, since the follow through that appears to be required for cleave wouldn't actually be part of the action, unless (again) the fighter is an idiot and continues to put his full wieght and force into blows that have not stuck anything even though he "hit" them dead center. At least in my games, we tend to think that Fighters, um, KNOW HOW TO WIELD A WEAPON, and therefore would not do that, as it would be almost suicidal to overbalance oneself like that in any real melee fight.
I realize that the FAQ says what it says, but in my games there is no FAQ, there is deductive reasoning on the part of the DM and not a FAQ.