I'm running downwind right behind another boat on the same tack — who keeps clear?

You do. With no overlap, Rule 12 applies: a boat clear astern must keep clear of a boat clear ahead on the same tack. The boat ahead is free to sail her course. Getting the overlap question right — overlapped or clear astern — is the whole of applying this rule, because the moment an overlap is established it becomes a Rule 11 problem instead.

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