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 $\mathrm{S}\mathrm{p}(2n,{\mathbb{F}}_q)$ 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 $\mathrm{S}\mathrm{p}(2n,{\mathbb{F}}_q)$, or of $\mathrm{S}\mathrm{O}(2n+1,{\mathbb{F}}_q)$, for any prime power $q$.