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So I know that a view key is not suffice to audit outgoing transactions.
Still, can't a person re-sign his outgoing transactions and produce a valid ring signature with the same key image to an auditor?

2

Yes, this is called a SpendProofV1 and is implemented by the official Monero wallet. See this answer: What is the "SpendProofV1..." or "OutProofV1..." in the details of a sent transaction in the GUI?

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