Background: One of the main objects of interest in the theory of $L$-functions is the root number, a complex number of modulus one which appears in the functional equation. In general, a root number can be attached to any complex Galois representation, or in the local case, any representation of the Weil(-Deligne) group, my case of interest. Deligne has a formula saying (roughly) that the root number of an orthogonal complex representation of the Weil group $W_k$ is its second Stiefel-Whitney class. I'm trying to understand the sense in which the standard constructions of Stiefel-Whitney classes in algebraic topology specialize to Deligne's case. (I'm mostly interested in the case where $k$ is nonarchimedean, though comments on the archimedean case are welcome as well.)
Question: In what sense does a complex orthogonal representation of the Weil group have Stiefel-Whitney classes?
Partial Progress: Let me explain some types of representations for which the construction is clear and some for which it is not.
Deligne first considers real representations. Here it is clear what to do. A real representation $\pi:W_k\to\text{GL}(V)$ factors through a discrete quotient $G$ of $W_k$. Functoriality of the classifying space gives a map $B\pi:BG\to B\text{GL}(V) \simeq BO(V)$, and we then pull back the universal Stiefel-Whitney classes along this map to produce classes $w_i(\pi)\in\text{H}^i(G,\mathbb Z/2\mathbb Z)$. That makes sense.
Now suppose $V$ is a complex vector space with a symmetric perfect pairing $V\otimes V\to\mathbb C$, and let $\pi:W_k\to O(V)$ be an orthogonal representation, which, as before, we can factor through a discrete quotient $G$. Such representations feature naturally in the Langlands program, though less naturally in algebraic topology as far as I know. (Beware that this orthogonal group is different from the compact one in the previous paragraph. The presence of the complex orthogonal group here has made googling impossible because the compact one is so much more common in topology.)
When $\pi$ is of Galois type, meaning that it factors through a finite quotient of $W_k$, I am again in a good situation: orthogonal complex representations of finite groups are (uniquely) realizable over $\mathbb R$ because they are unitarizable, and I can define the Stiefel-Whitney class of $\pi$ in this case as that of its Galois-descent to $\mathbb R$.
When $\pi$ is not of Galois type, this argument does not work because $\pi$ may not be realizable over $\mathbb R$. The simplest example is the unramified orthogonal representation with $$ \pi(\text{Frob}) = \begin{bmatrix} a & \\ & a^{-1} \end{bmatrix}, \qquad a\in\mathbb C $$ where $a+a^{-1}\notin\mathbb R$. So we can't define the Stiefel-Whitney classes of this representation by first descending to $\mathbb R$. This is the kind of representation for which I don't understand Deligne's assertion that it has Stiefel-Whitney classes. I had two ideas two account for the representations that are not realizable over $\mathbb R$, but both have problems.
First, we can always produce a real representation from a complex representation by restriction of scalars. But this construction does not seem to agree with the one that worked before, for representations of Galois type.
Second, we can use specific facts about the Weil group. Deligne can show that every (semisimple) complex orthogonal representation of the Weil group is a direct sum of a representation of Galois type and a representation of the form $W\oplus W^*$. You could hope to leverage this decomposition to get a definition. But Deligne's brief mention of the existence of Stiefel-Whitney classes makes me think that he's appealing to some more general construction of which I am unaware, and which would work for other discrete groups.