What does this equation mean?

Nothing about this phase disturbs conservation of energy or momentum, and it is worth saying explicitly why. $\Phi$[$\mathcal L_{\rm G}$] does not come from a new term added to any Hamiltonian; it comes from how already-unitary translation and boost operators compose. Each operator in the product T($\mathbf b$)G($\mathbf v$)T(-$\mathbf b$)G(-$\mathbf v$) conserves probability on its own, and the product of unitaries is unitary regardless of whether the factors commute, so the loop as a whole still conserves probability and leaves every expectation value of energy and momentum exactly where an ordinary, non-extended calculation would put it. What changes is not a conserved quantity but the bookkeeping of relative phase between branches that took different routes through the group — the same sense in which an Aharonov–Bohm loop conserves energy and momentum throughout while still leaving an interferometer with a phase shift at the end.