I have a CPTP quantum channel $\mathcal{E}$ that I've characterized by an operator sum representation $\{E_i\}$ for $i=1, \dots, m$ which acts on an input state like $$ \mathcal{E}(\rho) = \sum_{i=1}^m E_i \rho E_i^\dagger $$ with $\sum_i E_i^\dagger E_i = I$. I would like to know if there is an alternative Kraus representation with operators $\{F_j\}$ for $j=1, \dots, n$ that satisfies similar properties, $\mathcal{E}(\rho) =\sum_{j=1}^n F_j \rho F_j^\dagger$ and $\sum_{j=1}^n F_j^\dagger F_j=I$ but also that each new operator is proportional to a unitary, $$ F^\dagger_j F_j = F_j F_j^\dagger = c_j I $$ for some real $c_j$. For example a choice of dephasing channel satisfies this property since it can be defined either by the operators $\{\frac{1}{\sqrt{2}}I, \frac{1}{\sqrt{2}}Z\}$ or $\{|0\rangle\langle 0 |, |1\rangle \langle 1|\}$.

I do understand that the two channels must be related by some unitary transformation $U$ with, $E_i = \sum_j U_{ij} F_j$ but this seems only to constrain the problem for specific choices of $\{E_i\}$.