  We introduce a new model for random simplicial complexes which with high
probability generates a complex that has a simply-connected double cover. Hence
we develop a model for random simplicial complexes with fundamental group
$\mathbb{Z}/2\mathbb{Z}$. We establish results about the typical asymptotic
topology of these complexes. As a consequence we give bounds for the dimension
$d$ such that $\mathbb{Z}/2\mathbb{Z}$-equivariant maps from the double cover
to $\mathbb{R}^d$ have zeros with high probability, thus establishing a random
Borsuk--Ulam theorem. We apply this to derive a structural result for pairs of
non-adjacent cliques in Erd\H{o}s--R\'{e}nyi random graphs.
