  This study establishes consistency conditions and a general solution for a
coupled system that consists of five two-sided Sylvester-like tensor equations
in ten quaternion variables throughout the Einstein tensor product. Certain
specific cases are thus established. In a direct application, we investigate
certain necessary and sufficient conditions for the existence of an
$\eta$-Hermitian solution to five coupled two-sided Sylvester-like quaternion
tensor equations. Finally, we present an algorithm and a numerical example to
validate the main result.
