Place butter, garlic and cilantro in a food processor and process until smooth. Season with salt and pepper.
Heat olive oil in a large saute over high heat until almost smoking. Season char with salt and pepper and sear, skin side down, until golden brown, about 2 to 3 minutes. Turn over and continue cooking for 3 to 4 minutes or until just cooked through.
Grilled Tomato Broth:Preheat grill pan or grill. Slice tomatoes in half vertically, brush with oil, season with salt and pepper, grill until soft, and coarsely chop. Bring fish stock to a simmer, add the tomatoes and cook for 10 minutes.
Place the mixture in a food processor and process until smooth.
Put the mixture into a clean saucepan and bring to a boil.
Add the mussels and cook until all the mussels have opened, discarding any that do not.
Remove the mussels to a bowl.
Whisk a few tablespoons of the butter into the tomato broth and season with salt and pepper.
Add the mussels back to the broth. Stir in posole.
Place 1 char fillet in the center of each shallow bowl. Ladle sauce over and around the fish. Divide the mussels among the bowls and sprinkle with chopped cilantro.