  Real Clifford algebras play a fundamental role in the eight real
Altland-Zirnbauer symmetry classes and the classification tables of topological
phases. Here, we present another elegant realization of real Clifford algebras
in the $d$-dimensional spinless rectangular lattices with $\pi$ flux per
plaquette. Due to the $T$-invariant flux configuration, real Clifford algebras
are realized as projective symmetry algebras of lattice symmetries. Remarkably,
$d$ mod $8$ exactly corresponds to the eight Morita equivalence classes of real
Clifford algebras with eightfold Bott periodicity, resembling the eight real
Altland-Zirnbauer classes. The representation theory of Clifford algebras
determines the degree of degeneracy of band structures, both at generic $k$
points and at high-symmetry points of the Brillouin zone. Particularly, we
demonstrate that the large degeneracy at high-symmetry points offers a rich
resource for forming novel topological states by various dimerization patterns,
including a $3$D higher-order semimetal state with double-charged bulk nodal
loops and hinge modes, a $4$D nodal surface semimetal with $3$D surface
solid-ball zero modes, and $4$D M\"{o}bius topological insulators with a
eightfold surface nodal point or a fourfold surface nodal ring. Our theory can
be experimentally realized in artificial crystals by their engineerable
$\mathbb{Z}_2$ gauge fields and capability to simulate higher dimensional
systems.
