Police found the man's badly decomposed body inside the couple's Titahi Bay home on Sunday on the North Island of New Zealand.
Police have not confirmed the man's identity but the body is believed to be that of Debiprasad Majumdar.
New Zealand police have described the scene as "bizarre" and are now investigating the case of the death of the man, who was in his 50s and is believed to have been from India, New Zealand Herald reported.
According to neighbours, the couple had family in India, and had lived at the flat for about a year.
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The wife told neighbours the smell was from cat faeces, and sprayed Dettol outside the flat to cover the smell.
She was taken away by police after the discovery of her husband's body, but has not been charged.
Initial pathology tests were inconclusive and could not confirm if the death was a homicide, police said in a statement.
He said he first noticed the smell yesterday, and initially thought it was food that was thrown out to feed seagulls.
Rangihuna said the last time he saw the dead man was about three weeks ago. On Saturday he was having a cup of tea with a mate when he saw the woman walking outside his house.
Fellow neighbour Katerina Hill said the couple mostly kept to themselves, but the man used to go out for daily walks. She had not seen him for about four weeks, but had seen the partner right up until last week.