Brief Answer:
Unlikely due to cough. Needs work up

Detailed Answer:
Thank you for asking

I read your question and i understand your concern. Antibiotics are unlikely to cause fresh bleed per rectum so is bronchitis or cough. Even if bronchitis cause stool color change that will be black and tarry and that too in case only if you had blood in cough that you swallowed. No other thing explains direct connection.

Rest Mr Romick said very right, coughing may increase abdominal pressure and worsen the underlying bowel pathology like hemorrhoid that can cause this problem.

Talk to a general surgeon and let them have a look and assess you perianally and decide what is best for you. I hope it helps. take good care of yourself and don't forget to close the discussion please.

Regards
Khan

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