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1
Char chiles over a gas flame or under a broiler, turning occasionally, until blackened all over (this adds a smoky flavor and makes the skins easy to remove). Let cool. Using a small knife, scrape skin from chiles; discard. Coarsely chop chiles (for a milder salsa, discard seeds).
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2
Make a paste by mincing garlic and 1 teaspoon salt on a work surface, scraping with the side of knife occasionally. Add chiles and finely chop, then add mint and finely chop. Transfer to a bowl; stir in oil. Season with more salt, if desired. DO AHEAD:
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3
Place chicken and broth in a large heavy pot. Add water just to cover chicken. Bring to a simmer over high heat, skimming the surface occasionally (foam can add bitter flavors to soup; discard it as you skim). Reduce heat to medium-low; simmer, continuing to skim surface occasionally, for 5 minutes. Add corn cobs, halved onion, garlic, carrot, celery, peppercorns, bay leaves, and 2 teaspoons salt. Reduce heat to low and simmer until breast meat is just cooked through, 15-20 minutes.
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4
Transfer chicken to a cutting board; let cool slightly. Cut off breast meat; set aside (cooking breast meat longer will make it tough). Return chicken to pot. Simmer over low heat until legs and thighs are cooked through, about 20 minutes longer.
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5
Remove pot from heat. Transfer chicken to a platter; let cool slightly. Strain broth through a fine-mesh sieve into another large pot; discard solids. Shred all chicken meat, discarding skin and bones.
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6
Bring strained broth to a simmer. Add corn kernels and cook until heated through, about 3 minutes. Add red onion; cook until crisp-tender, about 5 minutes. Add chicken; simmer, stirring occasionally, until heated through, about 5 minutes. Season with salt.
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7
Divide soup among warm bowls. Garnish with tomato and tortilla chips. Squeeze lime wedges over; add wedges to soup. Serve, passing chile-mint salsa and sea salt alongside.
