Nodal lines inside the momentum space of three-dimensional crystalline solids are topologically stabilized by a π-flux of Berry phase. Nodal-line rings in PT-symmetric systems with negligible spin-orbit coupling (here described as "nodal class AI") can carry an additional "monopole charge", which further enhances their stability. Here, we relate two recent theoretical advancements in the study of band topology in nodal class AI. On one hand, cohomology classes of real vector bundles were used to relate the monopole charge of nodal-line rings to their linking with nodal lines formed among the occupied and among the unoccupied bands. On the other hand, homotopy studies revealed that the generalization of the Berry-phase quantization to the case of multiple band gaps defines a non-abelian topological charge, which governs the possible deformations of the nodal lines. In this work, we first present how to efficiently apply the non-abelian topological charge to study complicated nodal-line compositions. We then apply these methods to present an independent proof of the relation between the monopole charge and the linking structure, including all the fragile-topology exceptions. Finally, we show that the monopole charge flips sign when braided along a path with a non-trivial Berry phase. This facilitates non-abelian "braiding" of nodal-line rings inside the momentum space, that has not been previously reported. The geometric arguments presented in the main text are supplemented in the appendices with formal mathematical derivations.