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So, both Legion of the Damned and Inquisition allow you to take special "Other" designation detachments that are neither Primary nor Allied detachments.

Would it be legal to, for example, have a Dark Angels primary detachment, a Sisters of Battle allied detachment, and one each of LotD and Inquisition detachments?

As of right now, sure can. As long as the additional detachments do not generate and issue like Come the apocalypse, you are good to go.