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For people who have experience with TLA+ do you generate implementation code from the TLA+? Or is the idea to only use it for some formal design specifications?

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TLA+ is only for specification. In one of the papers on it Lamport mentions just putting a copy of the specification in a comment above the implementation. If you get a bug you just need to find where your implementation differs from your specification which he finds is usually pretty easy.

If your code doesn't have a specification (of course usually less precise), fixing bugs is incoherent. How can you say come behavior is a bug?

Yeah but why not generate code from your specification? Or create a model of the specification that you can interrogate in code?

You would need to include irrelevant details. TLA+ is an actual declarative language, so it's not executable, but it can be model-checked.

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