  We discuss a class of three-band non-Abelian topological insulators in three
dimensions which carry a single bulk Hopf index protected by spatiotemporal
($\mathcal{PT}$) inversion symmetry. These phases may also host subdimensional
topological invariants given by the Euler characteristic class, resulting in
real Hopf-Euler insulators. Such systems naturally realize helical nodal
structures in the 3D Brillouin zone, providing a physical manifestation of the
linking number described by the Hopf invariant. We show that, by opening a gap
between the valence bands of these systems, one finds a fully-gapped `flag'
phase, which displays a three-band multi-gap Pontryagin invariant. Unlike the
previously reported $\mathcal{PT}$-symmetric four-band real Hopf insulator,
which hosts a $\mathbb{Z} \oplus \mathbb{Z}$ invariant, these phases are not
unitarily equivalent to two copies of a complex two-band Hopf insulator. We
show that these uncharted phases can be obtained through dimensional extension
of two-dimensional Euler insulators, and that they support (1) an optical bulk
integrated circular shift effect quantized by the Hopf invariant, (2)
quantum-geometric breathing in the real space Wannier functions, and (3)
surface Euler topology on boundaries. Consequently, our findings pave a way for
novel experimental realizations of real-space quantum-geometry, as these
systems may be directly simulated by utilizing synthethic dimensions in
metamaterials or ultracold atoms.
