Let $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ be a CPTP map. Any such channel admits a Kraus decomposition of the form $$\Phi(X)=\sum_a A_a X A_a^\dagger,$$ for a set of operators $A_a\in\mathrm{Lin}(\mathcal X,\mathcal Y)$ satisfying $\sum_a A_a^\dagger A_a=I_{\mathcal X}$.
The standard way to prove this passes through the Choi representation $J(\Phi)$ of the channel, showing that CP is equivalent to $J(\Phi)$ being a positive operator, and therefore $J(\Phi)$ admits a spectral decomposition with positive eigenvalues, and finally realise that the eigenvectors of $J(\Phi)$ are essentially equivalent to the Kraus operators $A_a$ (upon some reinterpretation of the indices). This is shown for example at pag. 83 (theorem 2.22) of Watrous' TQI book, and in some form also in this other answer here, as well as in a slightly different formalism in this other answer of mine.
What puzzles me about this is the following. The components in the spectral decomposition of the Choi operator $J(\Phi)$ will also have to satisfy an additional property, one that I haven't seen discussed in this context: the orthogonality of the eigenvectors.
If $J(\Phi)=\sum_a v_a v_a^\dagger$, then we also know that the vectors $v_a$ are orthogonal. More specifically, we can always write $J(\Phi)=\sum_a p_a v_a v_a^\dagger$ for some $p_a\ge0$ and $\langle v_a,v_b\rangle=\delta_{ab}$. Remembering that here $v_a\in\mathcal Y\otimes\mathcal X$, these vectors are essentially the Kraus operators of the channel in the sense that $(v_a)_{\alpha i}=(A_a)_{\alpha i}$ (using greek and latin letters to denote indices in $\mathcal Y$ and $\mathcal X$, respectively).
The orthogonality of the $v_a$ is thus equivalent to the fact that Kraus operators must satisfy $$\operatorname{Tr}(A_a^\dagger A_b)\equiv \sum_{i\alpha}(A_a^*)_{\alpha i} (A_b)_{\alpha i}=p_a\delta_{ab}.\tag A$$
However, this property doesn't seem to be usually remarked. Moreover, people often refer to Kraus operators that do not satisfy this orthogonality condition. An example is the Kraus operators used for the dephasing channel in this answer.
The question is therefore as follows: should the property (A) be considered as a necessary condition for a set $\{A_a\}_a$ to be called a set of Kraus operators of a channel? Moreover, regardless of the terminology that one chooses to use, is there any advantage in choosing a "Kraus decomposition" for the channel that is made out of orthogonal operators, rather than non-orthogonal ones?