Heat oil in heavy large pot over medium heat.
Add leeks; sprinkle with salt and pepper and sauté until leeks begin to soften, stirring often, about 5 minutes.
Add mushrooms, garlic, and rosemary; increase heat to medium-high and sauté until mushrooms soften and begin to brown, stirring often, about 7 minutes.
Add tomatoes with juice; stir 1 minute.
Add barley and 4 cups broth; bring to boil.
Reduce heat to low, cover, and simmer until barley is almost tender, about 20 minutes.
Add kale; stir until wilted, about 1 minute.
Cover and simmer until kale and barley are tender, adding more broth by ¼ cupfuls as needed for desired stew consistency, about 10 minutes.
