  We present a framework to systematically address topological phases when
finer partitionings of bands are taken into account, rather than only
considering the two subspaces spanned by valence and conduction bands. Focusing
on $C_2\mathcal{T}$-symmetric systems that have gained recent attention, for
example in the context of layered van-der-Waals graphene heterostructures, we
relate these insights to homotopy groups of Grassmannians and flag varieties,
which in turn correspond to cohomology classes and Wilson-flow approaches. We
furthermore make use of a geometric construction, the so-called Pl\"ucker
embedding, to induce windings in the band structure necessary to facilitate
non-trivial topology. Specifically, this directly relates to the
parametrization of the Grassmannian, which describes partitioning of an
arbitrary band structure and is embedded in a better manageable exterior
product space. From a physical perspective, our construction encapsulates and
elucidates the concepts of fragile topological phases beyond symmetry
indicators as well as non-Abelian reciprocal braiding of band nodes that arises
when the multiple gaps are taken into account. The adopted geometric viewpoint
most importantly culminates in a direct and easily implementable method to
construct model Hamiltonians to study such phases, constituting a versatile
theoretical tool.
