  The realization graph $\mathcal{G}(d)$ of a degree sequence $d$ is the graph
whose vertices are labeled realizations of $d$, where edges join realizations
that differ by swapping a single pair of edges. Barrus [On realization graphs
of degree sequences, Discrete Mathematics, vol. 339 (2016), no. 8, pp.
2146-2152] characterized $d$ for which $\mathcal{G}(d)$ is triangle-free. Here,
for any $n \geq 4$, we describe a structure in realizations of $d$ that exactly
determines whether $G(d)$ has a clique of size $n$. As a consequence we
determine the degree sequences $d$ for which $\mathcal{G}(d)$ is a complete
graph on $n$ vertices.
