[tylerdurden]

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Please give me some historical examples from the capitalist era where the conditions you describe produced a standard currency (not sanctioned and upheld by government) that served effectively as the medium of exchange for a national economy.

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13th-Century northern Italy is full of examples. Some city-states (Genoa, e.g.) even had so little intervention that multiple metals circulated simultaneously (bimetallism is often described as "unstable" but that's only the case when government tries to enforce a fixed ratio of value between different metals).

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Then please explain to me how such a currency would be integrated into the contemporary financial system.

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I'm not sure how to answer the question. The "contemporary financial system" would adapt. That much is certain. I'm not particularly interested in predicting the particulars.