I've not noticed this happening if you give it the FEN each move. The alternative is just blindfold chess and very few humans can do that for long.

I haven't tried it myself, but people seem to report that the illegal moves surface eventually. It just takes longer: https://news.ycombinator.com/item?id=49720751

Nothing is forcing the LLM to play 'blind'. If it's smart, it should be able to create its own representation of the chess board and update it with every move, just like a human would. Any chess engine that's sensitive to how the moves are formatted is clearly not very capable.