  Blokhuis showed that all maximum cliques in Paley graphs of square order have
a subfield structure. Recently, it has been shown that in Peisert-type graphs,
all maximum cliques are affine subspaces, and yet some maximum cliques do not
arise from a subfield. In this paper, we investigate the existence of a clique
of size $\sqrt{q}$ with a subspace structure in pseudo-Paley graphs of order
$q$ from unions of semi-primitive cyclotomic classes. We show that such a
clique must have an equal contribution from each cyclotomic class and that most
such pseudo-Paley graphs do not admit such cliques, suggesting that the
Delsarte bound $\sqrt{q}$ on the clique number can be improved in general. We
also prove that generalized Peisert graphs are not isomorphic to Paley graphs
or Peisert graphs, confirming a conjecture of Mullin.
