  We develop a new technique to construct mutually unbiased tripartite
absolutely maximally entangled bases. We first explore the tripartite
absolutely maximally entangled bases and mutually unbiased bases in
$\mathbb{C}^{d} \otimes \mathbb{C}^{d} \otimes \mathbb{C}^{d}$ based on
mutually orthogonal Latin squares. Then we generalize the approach to the case
of $\mathbb{C}^{d_{1}} \otimes \mathbb{C}^{d_{2}} \otimes
\mathbb{C}^{d_{1}d_{2}}$ by mutually weak orthogonal Latin squares. The concise
direct constructions of mutually unbiased tripartite absolutely maximally
entangled bases are remarkably presented with generality. Detailed examples in
$\mathbb{C}^{3} \otimes \mathbb{C}^{3} \otimes \mathbb{C}^{3},$ $\mathbb{C}^{2}
\otimes \mathbb{C}^{2} \otimes \mathbb{C}^{4}$ and $\mathbb{C}^{2} \otimes
\mathbb{C}^{5} \otimes \mathbb{C}^{10}$ are provided to illustrate the
advantages of our approach.
