Here's how to make real carbonara without cream.

Carbonara
Ingredients:
250g Artisan Spaghetti or Bucatini
100g Pecorino Romano
100g Parmigiano Reggiano
300g-500g Thick cut bacon, Guanciale or Pancetta
3 egg yolks (large)
1 whole egg (large)
Fresh ground black pepper
Salt

Directions:
Cut bacon into strips.
Cook the bacon on low until fully cooked (~20min).
Add pepper and optionally some olive oil.
Grate pecorino and parmesan into powder.
Save a little for topping.
Add pepper to mixture.
Mix with egg yolks and whole egg.
Cook pasta al dente with 1 tbsp rock salt.
Put pasta in the pan and absorb bacon fat.
Take pan off heat completely.
Add egg mixture and pour a little pasta water on top.
Mix thoroughly and toss to make creamier.
Serve immediately.