Summer Walker Touring with Bronchitis, Refusing Medical Treatment

Summer Walker has what should be a show-stopping condition, but the R&B enchantress isn't stopping ANY performances ... because she knows fans paid good money to see her!!!

SW's currently overseas as the headliner for Australia and New Zealand's first-ever R&B and Soul music festival, 'Souled Out' ... but sources close to the "Over It" singer tell TMZ Hip Hop she came down with a case of bronchitis a few days ago.

We're told Summer doesn't believe in modern medicine, so she's been soothing her throat with a blend of herbs and teas in the meantime.

Summer battled through the 2nd date of the festival in Auckland, New Zealand, and the next one went down Friday in Melbourne, Australia.

We're told she's been on heavy vocal rest, meaning absolutely minimal speaking until her shows begin.