Real representations of finite symplectic groups over fields of characteristic two

We prove that when $q$ is a power of $2$, every complex irreducible representation of $\mathrm{Sp}(2n, \mathbb{F}_q)$ may be defined over the real numbers, that is, all Frobenius-Schur indicators are 1. We also obtain a generating function for the sum of the degrees of the unipotent characters of $\mathrm{Sp}(2n, \mathbb{F}_q)$, or of $\mathrm{SO}(2n+1, \mathbb{F}_q)$, for any prime power $q$.