ABSTRACT: We consider global issues with Relationalism: a first piece of Background Independence that resolves various Problem of Time facets and is locally implemented by Lie derivatives. Spacetime Relationalism and Configurational Relationalism are quite similar, though the second of these requires supplementing with a heterogeneous Temporal Relationalism. Avoiding both zeroes and Killing vectors is involved, as are some fibre bundle effects, including bibundles, monopoles on configuration space and the Gribov effect. Most of Relationalism’s globality follows from quotient configuration spaces being stratified manifolds. Nine distinct strategies for dealing with stratified manifolds are compared. Compactness and metric-space guarantees for nice (in particular Hausdorff) stratified manifolds are provided. Fibre bundles do not suffice for stratified manifolds, so general bundles, differential spaces, presheaves and sheaves are brought in instead. Hausdorff paracompact (HP) spaces continue to afford simplifications, with some support when local compactness and second-coutability apply. Relational Particle Mechanics, Gauge Theory and GR examples of these various global issues are provided.

[79 pages, including 25 figures. Second of a series of six Fundamental Physics and Applied Topology Articles.]