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So the Ruthlessness card says:

After you perform an attack that hits, you must choose 1 other ship at Range 1 of the defender (other than yourself). That ship suffers 1 damage.

And the tractor beam reference card (relevant section) says:

The first time a small ship receives a tractor beam token each round, the opposing player may choose one of the following effects:

- Perform a barrel roll using the [1 straight] maneuver template. The opposing player selects the direction of the barrel roll and the final position of the ship.

-Perform a boost using the [1 straight] maneuver template.

I'm trying to figure out whether the Ruthlessness damage triggers before or after the tractor beam move (or if the attacker can choose)...

I'm thinking that because (from the weapon card) it says "If this attack hits, the defender receives 1 tractor beam token." The sequence is hit-ruthlessness-move, but I could see it going the other way.

If it goes hit-move-ruthlessness it opens some interesting possibilities... probably still not worth it, but interesting

You're probably right, but it could end up being an order of operations gimmick. After the attack hits you do A. Ruthlessness and B. Give them a tractor token. So it could be you could pick which one happens first, as you are the attacker.

It's in the FAQ. ruthlessness triggers after the final position of the ship, after it boosts or barrel rolls from tractor beam.

It's good that the FAQ confirms that, but I'd say RAW also leads to the same conclusion. Tractor Beam resolves during the last part of the attack. Ruthlessness is "After you perform an attack that hits". That's after the attack is done resolving.