Let HI1×⋯×IN\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {H}^{I_{1}\times \cdots \times I_{N}}$$\end{document} be the set of the order N dimension I1×⋯×IN\documentclass[12pt]{minimal}
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\begin{document}$$I_{1}\times \cdots \times I_{N}$$\end{document} tensors over the real quaternion algebra H\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {H}$$\end{document}. For η∈{i,j,k}\documentclass[12pt]{minimal}
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\begin{document}$$\eta \in \{\mathbf {i},\mathbf {j},\mathbf {k}\}$$\end{document}, a quaternion tensor A∈HI1×⋯×IN×I1×⋯×IN\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}\in \mathbb {H}^{I_{1}\times \cdots \times I_{N}\times I_{1}\times \cdots \times I_{N}}$$\end{document} is said to be η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document}-Hermitian if A=Aη∗\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}=\mathcal {A}^{\eta *}$$\end{document}, where Aη∗=-ηA∗η,\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}^{\eta *}=-\eta \mathcal {A}^{*} \eta ,$$\end{document} and A∗\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}^{*}$$\end{document} stands for the conjugate transpose of A\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}$$\end{document}. In this paper, we establish some necessary and sufficient solvability conditions for a system of quaternary-coupled Sylvester-type quaternion tensor equations. We give an expression of the general solution to this system when it is solvable. As an application of this system, we provide some necessary and sufficient conditions for the existence of a solution to the system of coupled Sylvester-type quaternion tensor equations involving η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document}-Hermicity: A1∗NX-Y∗NB1=C1,A2∗NX-Z∗NB2=C2,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{c} \mathcal {A}_{1}*_{N}\mathcal {X}-\mathcal {Y}*_{N}\mathcal {B}_{1}=\mathcal {C}_{1},\\ \mathcal {A}_{2}*_{N}\mathcal {X}-\mathcal {Z}*_{N}\mathcal {B}_{2}=\mathcal {C}_{2}, \end{array}\right. \end{aligned}$$\end{document}where the operation ∗N\documentclass[12pt]{minimal}
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\begin{document}$$*_{N}$$\end{document} is the Einstein product, Ai,Bi,\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}_{i},\mathcal {B}_{i},$$\end{document} and Ci(i=1,2)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {C}_{i}~(i=1,2)$$\end{document} are given tensors, X,Y,\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {X},\mathcal {Y},$$\end{document} and Z\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {Z}$$\end{document} are unknowns and Z\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {Z}$$\end{document} is η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document}-Hermitian. We also present an expression of the general solution to this system when the solvability conditions are met. Some algorithms and numerical examples are presented to illustrate the results of this paper.
