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Here is one instance, aalthough not with a "classical" choice principle.

Namely, your principle TC implies the rigid relation principle RR, a weak choice principle introduced by Justin Palumbo and myself in this paper:

• Joel David Hamkins and Justin Palumbo, The rigid relation principle, a new weak choice principle, Math Logic Quarterly, 58 (6):394-398 (2012), DOI:10.1002/malq.201100081, arXiv:1106.4635.

The rigid relation principle is the assertion that every set admits a rigid binary relation. This is true under TC, since $\langle X,\in\rangle$ is rigid whenever $X$ is a transitive set, and so every set that is bijective with a transitive set admits a rigid binary relation.

Meanwhile, Justin and I proved that RR is strictly intermediate between ZF and ZFC.

Here is one instance, although not with a "classical" choice principle.

Namely, your principle TC implies the rigid relation principle RR, a weak choice principle introduced by Justin Palumbo and myself in this paper:

• Joel David Hamkins and Justin Palumbo, The rigid relation principle, a new weak choice principle, Math Logic Quarterly, 58 (6):394-398 (2012), DOI:10.1002/malq.201100081, arXiv:1106.4635.

The rigid relation principle is the assertion that every set admits a rigid binary relation. This is true under TC, since $\langle X,\in\rangle$ is rigid whenever $X$ is a transitive set, and so every set that is bijective with a transitive set admits a rigid binary relation.

Meanwhile, Justin and I proved that RR is strictly intermediate between ZF and ZFC.

Your principle TC implies the rigid relation principle RR, a weak choice principle introduced by Justin Palumbo and myself in this paper:

• Joel David Hamkins and Justin Palumbo, The rigid relation principle, a new weak choice principle, Math Logic Quarterly, 58 (6):394-398 (2012), DOI:10.1002/malq.201100081, arXiv:1106.4635.

The rigid relation principle is the assertion that every set admits a rigid binary relation. This is true under TC, since $\langle X,\in\rangle$ is rigid whenever $X$ is a transitive set, and so every set that is bijective with a transitive set admits a rigid binary relation.

Meanwhile, Justin and I proved that RR is strictly intermediate between ZF and ZFC.

Here is one instance, aalthough not with a "classical" choice principle.

Namely, your principle TC implies the rigid relation principle RR, a weak choice principle introduced by Justin Palumbo and myself in this paper:

• Joel David Hamkins and Justin Palumbo, The rigid relation principle, a new weak choice principle, Math Logic Quarterly, 58 (6):394-398 (2012), DOI:10.1002/malq.201100081, arXiv:1106.4635.

The rigid relation principle is the assertion that every set admits a rigid binary relation. This is true under TC, since $\langle X,\in\rangle$ is rigid whenever $X$ is a transitive set, and so every set that is bijective with a transitive set admits a rigid binary relation.

Meanwhile, Justin and I proved that RR is strictly intermediate between ZF and ZFC.

This is a partial answer, but let me mention that the transitive choice principle TC implies the rigid relation principle RR, a weak choice principle introduced by Justin Palumbo and myself in this paper:

• Joel David Hamkins and Justin Palumbo, The rigid relation principle, a new weak choice principle, Math Logic Quarterly, 58 (6):394-398 (2012), DOI:10.1002/malq.201100081, arXiv:1106.4635.

The rigid relation principle is the assertion that every set admits a rigid binary relation. This is true under TC, since $\langle X,\in\rangle$ is rigid whenever $X$ is a transitive set.

Meanwhile, Justin and I proved that RR is strictly intermediate between ZF and ZFC.

Your principle TC implies the rigid relation principle RR, a weak choice principle introduced by Justin Palumbo and myself in this paper:

• Joel David Hamkins and Justin Palumbo, The rigid relation principle, a new weak choice principle, Math Logic Quarterly, 58 (6):394-398 (2012), DOI:10.1002/malq.201100081, arXiv:1106.4635.

The rigid relation principle is the assertion that every set admits a rigid binary relation. This is true under TC, since $\langle X,\in\rangle$ is rigid whenever $X$ is a transitive set, and so every set that is bijective with a transitive set admits a rigid binary relation.

Meanwhile, Justin and I proved that RR is strictly intermediate between ZF and ZFC.