This paper studies the two-way wiretap channel (TW-WC) with an external eavesdropper under strong one-sided (mixed) secrecy: only User~1's message is required to be secure from the eavesdropper, while no secrecy constraint is imposed on User~2's message. Secrecy is measured by the information leakage of User~1's message to the eavesdropper. Applying a non-adaptive construction and a one-sided reduction of the key-exchange construction, we obtain exponential error and leakage bounds for every fixed number of adaptive rounds. A quantitative one-time-pad argument propagates the leakage of the key used in the next round, and an explicit padding construction removes the initialization-rate loss for all sufficiently large blocklengths. We derive the corresponding strong mixed-secrecy achievable regions. The non-adaptive construction recovers the previously reported weak one-sided single-letter inner bound under the stronger secrecy criterion. The overall adaptive achievable region includes the non-adaptive construction as a fallback and also contains a key-exchange subregion that can be strictly larger.

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