Unirational del Pezzo surfaces of degree one

We construct explicit unirational del Pezzo surfaces of degree $1$ with arithmetic Picard rank one over $\mathbb{Q}$, ${\mathbb{F}}_5$, and $\mathbb{C}(t)$. Moreover, we prove that every smooth real geometrically rational surface is unirational over $\mathbb{R}$ if and only if it has a real point.