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If I have an Iron Warriors patrol detachment and a Chaos Daemons patrol detachment all mark of Nurgle and Nurgle daemons can I still use Iron Warrior's stratagems? A slight explanation of how allegiance works.

You get both daemon and chaos marine stratagems. If you have appropriate keyworded detachments it works.

What's more if keywords on stratagems match you can use it even for other book's units! Albeit rare and deepstrike one on daemons iirc got　faqed that doesn't work

To clarify, you cannot use Chaos Daemon Stratagems on Heretic Astartes units even if they have the appropriate keywords due to an FAQ clarification. You could do the opposite.

More importantly, any keyword-based abilities from either type of unit works on the others as long as they have the correct keywords.

Including psychic powers?

Stratagems arent psychic powers. Those work fine, if the keywords match.