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I know this question is probably an obvious "No they can't! You're an idiot! *slap*", but I want to check:

You have a Marine army with Legion of the Damned. That army is allied to another Marine army, and that army contains Tiggy. Am I correct in assuming that Tiggy's re-roll for Reserves cannot be used with the Legion of the Damned in the main army because, although part of the same army, they are in different detachments and Tiggy's rules specifically mentioned people from the same detachment, not just the army as a whole?

Pretty sure you got it right.