Place anchos and guajillos in Dutch oven and cook over medium heat, stirring often, until fragrant and darkened slightly but not smoking, 3 to 5 minutes.
Immediately transfer anchos and guajillos to bowl and cover with hot water. Let sit until chiles are soft and pliable, about 5 minutes.
Drain anchos and guajillos and combine with tomatoes and their juice, 1 cup water, chipotle(s), soy sauce, and 1½ teaspoons salt in blender. Process until smooth, 1 to 2 minutes; set aside.
Heat oil in now-empty Dutch oven over medium-high heat until shimmering. Add onion, poblano, and remaining ¾ teaspoon salt. Cook, stirring occasionally, until onion begins to brown, 3 to 5 minutes.
Stir in tomato paste, garlic, cumin, and oregano and cook until tomato paste darkens, 1 to 2 minutes.
Stir in pinto, black, and kidney beans; barley; mushrooms; chile puree; and 2½ cups water. Bring to boil.
Reduce heat to medium-low and simmer, stirring occasionally, until barley is tender, 35 to 45 minutes.
Let sit off heat for 10 minutes (chili will continue to thicken as it sits).
Season with salt to taste. Stir in cilantro and serve.
