The atom-free socks symmetric system

Let $P$ be the finite partial functions on $(\omega \times 2\times \omega )\times \omega$ with binary values. The coordinate $(n,i,j)$ adds a real $r_{n,i,j}$. For fixed $n,i$, put $R_{n,i}=\{r_{n,i,j}:j\in \omega \}$ and $P_n=\{R_{n,0},R_{n,1}\}$.

Automorphisms preserve $n$, may swap the two $i$-blocks independently for each $n$, and permute the $j$-coordinates within blocks. The normal filter is generated by finite pointwise coordinate stabilizers. Each unordered $P_n$ and the graph $n\mapsto P_n$ have empty support, while an individual $R_{n,i}$ requires a coordinate in the $n$-block. The symmetric interpretation is a ZF model by Hereditarily symmetric interpretations form a transitive ZF model. This is the pure-set analogue of socks; no atoms or AC occur.