No, you can't eat cheese, mayo or bread, but eggs are ok. Maybe if you do homemade Mayo, but don't know the ingredients so can't say for sure.
It's a very strict diet but after a while you get used to it. I eat a lot of fish, chicken, meat (not pork), eggs, broccoli, avocados, extra virgin olive oil, coconut oil and some butter. I think that's it.
I have tried to add some coconut flour and buckwheat but it gives me a lot of gas.