Meaning
Rectangular compatibility means that a gauge comparison and a regional restriction can both be posed on the same valid data.
For a gauge tuple and a spacetime tuple, every needed presentation, restriction, marking, coefficient line, filling, seam, operator domain, and background must be meaningful on their common support. Compatibility must survive deletion or repetition of indices.
Example (illustrative)
To compare “change gauge, then restrict region” with “restrict region, then change gauge,” both routes must be defined on the same relevant support. If one route uses a collar excluded by the other, the proposed rectangle is not yet compatible.
Scope and limits
Well-defined routes need not agree. Their physical comparison cell and any residual defect are additional data.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 5.3.