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Bad smell during breathing could be due to various reasons. Let us see those one by one.
Malignancy: an intranasal (inside the nose) or intraoral (inside the mouth) malignancy can produce foul smell, that can be perceived as faeces smell by the nose. Malignancy can be painless and without any symptoms. Malignancies of paranasal air sinuses also produce foul odor.
Dental infection: a tooth decay can produce bad smell. Dental infection usually produces significant pain. Food particles get collected inside the decayed tooth and hence cause the bad smell.
Sinusitis: Chronic sinusitis can cause constant post nasal drip and purulent discharge (pus discharge). This results in foul odor. Maxillary sinusitis commonly produces foul odor. Sinusitis will have other symptoms like headache, facial pain, nasal blockage, post nasal drip, etc.
Foreign body inside nose: A foreign body which has got stuck inside nose without being removed for many days will produce purulent discharge and smell. The discharge will be from one side of the nose. It is common among kids. Foreign body produces nasal block.
Atrophic rhinitis: It is a disease where the normal space inside the nose gets increased due to destruction of the anatomy. Also the condition causes merciless anosmia (loss of sensation of smell). The sufferer cannot perceive any smell but the people around him can get the extreme foul odor emanating from the patients nose. Atrophic rhinitis also presents with crusting inside nose and nose block.
Maggots in nose: Maggots are larval stage of fly (chrysomia). Maggots usually occur in association with atrophic rhinitis. Fly lays its eggs inside nose leading to maggots inside nose. They are attracted to the foul odor and they themselves aggravate the odor.
Tonsillitis: Pus accumulated over the tonsil can cause bad smell. Tonsillar enlargement with infection and consequent foul breath is common in many children. Other symptoms of tonsillitis include fever, throat pain, enlarged lymph nodes of neck, difficulty and pain while swallowing.
Nasal diphtheria can produce foul odor. Diphtheria produces a membrane over the affected mucosa.
Rhinitis caseosa: it is a destructive disease producing foul odor. It was thought to be a fungal disease, exact etiology is still not clear.
Tuberculosis: TB of nasal mucosa produces granulomatous inflammation and caseating necrosis, consequent death of tissues and foul smell.
Hope I have cleared your query. Further evaluation and clinical examination is required.
Regarding follow up:
For further follow up, consult an ENT otolaryngologist online.---> https://www.icliniq.com/ask-a-doctor-online/ENT-Otolaryngologist