What are examples of spoiled food that is part of a culture's cuisine?
The difference between "rot" and "ferment" is extremely thin. The use of bacteria and fungus to partially break down foods is as old as any cultivation process. "Rotten" is just the word we use when the microorganisms have done something we don't like.

Fermented (rotten) foods are incredibly common, from cultured butter, ghee, buttermilk and yogurt to cheese, wine, soy sauce and beer. Not to mention bread, Tabasco, Sriracha, kim chi and sauerkraut, or heck, salami, chorizo, kombucha and fish sauce. Wikipedia has a huge list. We use bacteria like Lactobacillus, yeast like Saccharomyces cerevisiae, mold like Penicillium roqueforti and thousands more. Some are cultured specifically for their task, and others occur naturally and are allowed to do their business without interruption.

A few that come to mind that are less commonly considered and fairly interesting to me:

Bonito - Fish are dried, smoked and inoculated with mold (Aspergillus glaucus). A staple flavoring in Japanese cooking.

Kumis - A carbonated, fermented milk product. Lightly alcoholic, and traditionally made from mare's milk, and fermented with wild yeast and bacteria. Quite the beverage, I hear.