Meaning
A global admissible realization is a coherent choice of local realizations and comparisons throughout the selected indexing category.
It is a section of the admissible Grassmannian fibered object: one admissible object at each index, with the required comparisons along arrows. Exact composition gives a strict section. Specified unitary or correspondence coherence gives a twisted section.
Example (illustrative)
With stages α→β→γ, the chosen direct comparison from α to γ must agree with the composite through β, exactly or through the prescribed comparison cell. Choosing an object at each stage without this relation does not define a section.
Scope and limits
Separate nonempty fibers do not imply a global realization. Persistent failure of the triangle or higher coherence equations can obstruct one.
Source
GAT I — Grassmannian Gluing, Spectral Trichotomy, and Conditional Sectorial Type II. GAT I, Definition 3.3.