Are there any major commodity markets with a public order book?
Major trading exchanges have order books that anyone can see, if they're willing to pay for the service. So in that sense, they are available to the public. For example, anyone can subscribe via a broker to the London Stock Exchange Level 2 data, which shows them the live order books for listed instruments.

Very few have "secret" order books. I think you mean dark pools; they do exist too, and wikipedia has a list.