For a filtration
each stage adds local data and coherence. The filler space at a stage is the homotopy fiber of the map from the next reconstruction object to the current one.
Example. In a crossed-module cocycle, one may first add missing object data, then arrow data, then 2-morphisms, checking after each addition whether the partial field has a coherent continuation.
Firewall. Nonempty or connected filler spaces do not imply reconstruction. Contractibility is the relevant uniqueness condition.
Source
Admissible Witness Descent for Higher Gauge Fields, public website edition v1.0. Not peer reviewed.