The way LLMs work is they output probabilities for every _token_, so you don't really need to backtrack you can just always pick a token that matches the provided grammar.

That said, you might want to do something like (backtracking) beam-search which uses various heuristics to simultaneously explore multiple different paths because the semantic information may not be front-loaded, i.e. let's say we had a grammar that had a key "healthy" with values "very_unhealthy" or "moderately_healthy." For broccoli, the LLM might intend to say "very_healthy" and choose "very" but then be pigeonholed into saying "very_unhealthy" because it's the only valid completion according to the grammar.

That said, there are a lot of shortcuts you can take to make this fairly efficient thanks to the autoregressive nature of (most modern) LLMs. You only need to regenerate / recompute from where you want to backtrack from.