  We study the realization spaces of $10_3$ line configurations. Answering a
question posed by Sturmfels in 1991, we use elliptic surface techniques to show
that realizations over $\mathbb{Q}$ are dense in those over $\mathbb{R}$ for
all $10_3$ configurations. We find that for exactly four of the ten
configurations, the realization space admits a compactification by a K3
surface. We show that these have Picard number 20 and compute their
discriminants. Finally, we use geometric invariant theory to give an elegant
interpretation of these K3 surfaces as moduli spaces.
