  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 $q$.
