In a large pot over medium-high heat, heat 1 tbsp. oil. Add barley and cook, stirring occasionally, until toasted and fragrant, about 5 minutes. Transfer barley to a small bowl.

In same pot over medium heat, heat remaining 2 tbsp. oil. Add chard stems, onion, fennel, and garlic; season with salt and pepper. Cook, stirring occasionally, until just softened, about 5 minutes. Add tomato paste and cook, stirring frequently, until brick red, about 1 minute. Add coriander, cumin, allspice, and ginger. Cook, stirring, until tomato paste is caramelized and spices are fragrant, 3 to 5 minutes.

Add broth, bay leaves (if using), and 1 tbsp. salt; cover and bring to a boil. Add barley, reduce to a simmer, and cook 15 minutes. Add potatoes and cook, stirring occasionally, until barley and potatoes are tender, 12 to 15 minutes.

Add chard leaves, beans, vinegar, and sugar. Cook, stirring frequently, until chard leaves are wilted and beans are warmed through, about 2 minutes. Stir in ½ c. dill and ½ c. mint; season with salt and pepper. Discard bay leaves (if using).

Divide stew among bowls and top with remaining ½ c. dill and ½ c. mint.