  We prove the existence of two non-isomorphic number fields $K$ and $L$ such
that the maximal two-step nilpotent quotients of their absolute Galois groups
are isomorphic. In particular, one may take $K$ and $L$ to be any of the
imaginary quadratic number fields of discriminant -11, -19, -43, -67, -163.
Furthermore, we give an explicit combinatorial description of these Galois
groups in terms of a generalization of the Rado graph. A critical ingredient in
our proofs is the back-and-forth method from model theory.
