Thanks for the responses! I think my table has seen player conflict before, but it has never resulted in violence.
Here, I definitely did not announce any offscreen badness to put the player conflict in perspective, and so the player conflict rapidly overtook all other concerns. That strikes me as solid practical advice.

I do think I will return to our normal program of swords and sorcery for a time. It was really myself and the Battlebabe who were keen on playing Apocalypse World, so I'm not sure the others really had a strong feeling about what their AW experience should be about.

My AW session last night ended with the Brainer going aggro on the Faceless, the Quarantine shooting the Brainer, the Brainer pulling in-brain puppet strings on the Quarantine to make him attack the Faceless, the Faceless blowing the Quarantine away with an RPG, and the Battlebabe ending the Faceless as retribution.

That was not what I was hoping for? They certainly were all invested in the outcome, so I suppose I succeeded at capturing everyone's attention, but there were some sour feelings in the middle of this action so in general the scene is something I hope to avoid in the future. This is a stable group that has been gaming together for six years or so, though this was only our third session of AW.

I fear I precipitated these events as MC. The Faceless failed a Norman roll, so I told him his mask wanted him to hold the hardhold's water purifier hostage. He was perhaps a bit overconfident from having completely mowed over whatever obstacles I had thrown at him before by violence, so he did not attempt to seek healing for his wounds before laying claim to the hardhold's water purifier, which was an action designed to spark a response from pretty much every character, MC or player, that we had introduced.

Was giving the Faceless a motivation to take something designed to elicit so many hard moves staying true to my principles of being a fan of the characters? I feel like the Norman MC move was a valid hard move to make, but I'm not certain. I think, in retrospect, I probably would have still made that same hard move but I should have told more characters the consequences and asked in the build up to the climactic showdown, as I did not gently remind my players they had some alternatives to aggressive moves.

As an aside, I will note that I think my players used violence a lot more than I had expected, though by the third session they were hip to the fact that the number of souls in their little hardhold was rapidly dwindling.

Any MC advice, from the book or otherwise?

I have never played in a game where PCs were really out to kill each other, trading blows back & forth, or waging a multi-session campaign against each other.

I have engaged in a little bit of serious character conflict, however, though it never turned violent. The climactic session in the last campaign I played featured a cleric of the Broken God (Domain of the maimed, deformed, disfigured, diseased, deranged) who claimed the mantle of the Broken God after freeing him from his bondage. I, playing as wizard of the good persuasion, attempted to prevent this particular vacancy in the pantheon from being filled. I fictionally wasn't positioned to do anything about it apart from casting a spell, though, so I had to succeed on my Perfect Summons but I failed the roll. Once the cleric had achieved Godhood, I did not have any moves left to me but Ritual, which I triggered. GM ruled I needed the tears of the God of Healing to fix my deranged cleric-cum-god friend, so I Perfect Summons-ed the God of Healing to beseech him for his tears. Ultimately, the GM ruled the God of Healing wasn't going to accede to my request so I had my wizard dedicate himself to the service of the God of Healing to thwart the effect of the new Broken God on the world, and chose the new class of cleric.

We, of course, retired those characters at that point.

All this is to say the GM principle "Be a fan of the characters" can be tricky to fulfill when the characters are pulling the GM in different directions. I myself would hope never to GM a true arena-style PVP session.