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This paper determines the homogeneous identities of degree ≤ 7 satisfied by the teary anti-commutator [a, b, c] :＝ abc+acb+bac+bca+cab+cba in triple systems satisfying either total associativity (abc)de ≡ a(bcd)e ≡ ab(cde)or partial associativity (abc)de + a(bcd)e + ab(cde) ≡ 0. These identities define new teary versions of Jordan algebras different from Jordan triple systems.