We investigate how an affine connection ∇\documentclass[12pt]{minimal}
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\begin{document}$$\nabla $$\end{document} that generally admits torsion interacts with both g and L on an almost (para-)Hermitian manifold (M,g,L)\documentclass[12pt]{minimal}
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\begin{document}$$(\mathfrak {M},g,L)$$\end{document}, where L denotes either an almost complex structure J with J2=-id\documentclass[12pt]{minimal}
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\begin{document}$$J^{2}=-\hbox {id}$$\end{document} or an almost para-complex structure K with K2=id\documentclass[12pt]{minimal}
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\begin{document}$$K^{2}=\hbox {id}$$\end{document}. We show that ∇\documentclass[12pt]{minimal}
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\begin{document}$$\nabla $$\end{document} becomes (para-)holomorphic and L becomes integrable if and only if the pair (∇,L)\documentclass[12pt]{minimal}
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\begin{document}$$ (\nabla ,L)$$\end{document} satisfies a torsion coupling condition. We investigate (para-)Hermitian manifolds M\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {M}$$\end{document} in which this torsion coupling condition is satisfied by the following four connections (all possibly carrying torsion): ∇,∇L,∇∗,\documentclass[12pt]{minimal}
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\begin{document}$$\nabla ,\nabla ^{L},\nabla ^{*},$$\end{document} and ∇†=∇∗L=∇L∗\documentclass[12pt]{minimal}
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\begin{document}$$\nabla ^\dagger = \nabla ^{*L}=\nabla ^{L*}$$\end{document}, where ∇L\documentclass[12pt]{minimal}
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\begin{document}$$\nabla ^{L}$$\end{document} and ∇∗\documentclass[12pt]{minimal}
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\begin{document}$$\nabla ^{*}$$\end{document} are, respectively, L-conjugate and g-conjugate transformations of ∇\documentclass[12pt]{minimal}
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\begin{document}$$\nabla $$\end{document}. This leads to the following special cases (where T stands for torsion): (i) the case of T=T∗,TL=T†\documentclass[12pt]{minimal}
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\begin{document}$$T = T^*, T^L = T^\dagger $$\end{document}, for which all four connections are Codazzi-coupled to g, but dω≠0\documentclass[12pt]{minimal}
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\begin{document}$$d\omega \ne 0$$\end{document}, whence M\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {M}$$\end{document} is called Codazzi-(para-)Hermitian; (ii) the case of T=-T†,TL=-T∗\documentclass[12pt]{minimal}
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\begin{document}$$T = - T^{\dagger }, T^L = - T^{*}$$\end{document}, for which dω=0\documentclass[12pt]{minimal}
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\begin{document}$$d \omega = 0$$\end{document}, i.e., the manifold M\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {M}$$\end{document} becomes (para-)Kähler. In the latter case, quadruples of (para-)holomorphic connections all with non-vanishing torsions may exist in (para-)Kähler manifolds, complementing the result of Fei and Zhang (Results Math 72:2037–2056, 2017) showing the existence of pairs of torsion-free connections, each Codazzi-coupled to both g and L, in Codazzi-(para-)Kähler manifolds.
