In a very large (8-10 quart) stockpot set over medium high, combine the fish bones, onion, 5 of the garlic cloves and 3 ½ quarts of water. Bring the mixture to a boil, skim off all the grayish foam that rises during the first few minutes, then lower the heat, cover, and simmer for 30 minutes. Strain through a medium mesh strainer into a large bowl; discard the solids.
Place the tomatoes and the remaining garlic cloves in a blender or food processor and process until smooth. Measure 1 tablespoon of the oil in the stockpot over medium-high and when hot add the tomato puree (it should sizzle immediately). Cook, stirring, until it thickens to the consistency of tomato paste, about 15 minutes. Stir in the beer, the fish broth and the chayote. Simmer until the chayote is tender, about 7 minutes, and season with salt (usually 4 teaspoons).
Add the remaining 2 tablespoons of the oil in a medium (8-inch) skillet set over medium. When hot, add the chile rings and toast until just fragrant, about 15 seconds. Remove the chiles with a slotted spoon and drain on paper towels. Just before serving the soup, stir in the hoja santa (or fennel tops) and divide it among 6 bowls. Top each serving with a spoonful of chiles and a few cubes of cheese, and serve.