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So in order to take a relic, you must have a warlord that is of that faction. However say you have another detachment of a different faction, eg you had thousand sons warlord but also a deathguard detachment. This grants you access to the death guard stratagems and you could take a thousand sons relic. Could you use the gifts of decay stratagem to take a death guard relic for one command point? The stratagem says your army can have one extra relic of decay. You had 0 to start with but one extra would mean 1 relic right?

Yes, as answered in the Codex Death Guard FAQ.

Just for clarification:

If I have one detachment of Word Bearers (this one has the warlord), one detachmen of Death Guard and one detachment of Thousand Sons, can I spent 3 CP to have a total of 4 relics in my army, two for CSM and one each for TS and DG?

Yes. Just as you listed by using the appropriate Strategems.

do note however that each relic has to be on a different character (In this scenario, you would need 5 characters for the five different relics)