In a large saucepan, heat 2 tablespoons olive oil over medium heat. Add the onion, season with salt and cook, stirring occasionally, until the onions are softened and translucent, 5 to 6 minutes. Stir in the garlic and ginger until fragrant, about 1 minute. Add the mushrooms, season with salt and plenty of pepper, and cook, stirring occasionally and adding 1 to 2 tablespoons of water if the mushrooms begin to stick to the bottom of the pan, until their liquid is released, 5 to 7 minutes. Add the barley and cook, stirring occasionally, 1 minute, then stir in the beef broth and 2 cups water. Cover and bring to a boil, then lower the heat to a simmer and cook, covered, until the barley is tender, 30 to 40 minutes; add the bok choy for the last 10 minutes of cooking—it should be crisp-tender. Season with the lime juice to taste, 1 tablespoon at a time. Ladle into bowls and top with the cilantro.