If a ship with the “Two hits from the same shoot action are required to eliminate one of this ship’s masts.” ability is hit by something with “Once per turn when this ship hits an enemy ship, also eliminate one crew from that ship.” does it still eliminate a crew if it only scores one successful hit, so not causing a mast to be lost?
Likewise, would something like La Belle Etoile still cause a ship with “Two hits from the same shoot action are required to eliminate one of this ship’s masts.” to still lose something on every successful hit regardless of how many masts are actually eliminated?
Yes to both of these. Those abilities are triggered by hitting a ship, not eliminating a mast.
Exactly – Both of those abilities work in full, because hits and mast eliminations are separate things that may or may not trigger other effects depending on their specific wording.
If you also throw in the ability that eliminates two masts with a single hit, you end up with a sequence something like this:
First Hit – Eliminates a crew, but no mast damage.
Second Hit – Eliminates two masts.
Third Hit – No damage.
Fourth Hit – Eliminates two masts.
Fifth Hit – Etc. (alternating)
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A ship like Asesino de la Nave (or La Belle Etoile) will still destroy cargo with every single hit, even if only every second hit eliminates a mast on the target.