In Deligne's article 'Les constantes des equations fonctionelles des fonctions L' http://publications.ias.edu/sites/default/files/Number20.pdf"Les constantes des equations fonctionelles des fonctions L", we find the following claim:
Proposition 8.9 (ibid.): Suppose $(V, \rho, N )$ and $(V', \rho', N')$ are two Frobenius semisimple, $\ell$-adic and $\ell'$-adic representations of the Weil-Deligne group $' W_K $ of a (non-archimedean ) local field $ K$, respectively. Let $\mathbf{M}_{\bullet}$ and $\mathbf{M'}_{\bullet}$ be the (monodromy) filtrations induced by $N $ and $N'$ on $ V$ and $V'$, respectively. Then $(V, \rho, N )$ and $(V', \rho', N')$ are compatible (in the sense of ibid. 8.7) if and only if the characters, of the representations induced by $\rho$ and $\rho '$ on the graded parts of $\mathbf{M}_{\bullet}$ and $\mathbf{M'}_{\bullet}$ coincide and has values in $\mathbb{Q}$.
Deligne states this as an exercise. Does anyone know a rigorous proof or any reference?
Of course it is clear that if the characters of the representations (as in the proposition) induced by $\rho $ and $ \rho'$ on the graded parts of the filtrations $\mathbf{M}_{\bullet}$ and $\mathbf{M'}_{\bullet}$, coincide and has values in $\mathbb{Q}$, then the character of $\rho $ and $ \rho'$ are same and hence due to semi-simplicity, $\rho $ and $\rho'$ are isomorphic (after base extension to a common algebraic closure $\Omega \supset \mathbb{Q}_{\ell}, \mathbb{Q}_{\ell '} $) as representations of the Weil group $ W_K $. But what is not clear to me is how to find an isomorphism $ f $ between $ \rho$ and $ \rho' $ ( after base extension $\Omega $), such that
$$ f \circ N = N' \circ f \quad \text{(after base extension to $\Omega $) } \;? $$
