The incident happened on Monday evening in the district just north of Gare du Nord known as "Little Jaffna" due to the high number of Sri Lankan shops and restaurants there.
When bystanders saw the theft taking place, about a dozen of them chased the 26-year-old down the street and when they caught him began beating him before leaving him on the footpath and making off, Le Parisien newspaper reported.
Locals alerted the police about what they thought was a brawl and when officers arrived on the scene at Rue Cail in the 10th arrondissement, they found the young man lying semi-conscious and bleeding on the ground.
Next to him lay the bag he had grabbed from a Sri Lankan tourist a few minutes before.
The police officers quickly saw that he had two knife wounds to the stomach and called for an ambulance. The man was brought to hospital in a critical condition, according to Le Parisien.
The paper said that police efforts to find eyewitnesses to the incident drew a blank.
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Tourists, particularly foreign ones, are frequently targeted by thieves and pickpockets in Paris. But it is rare for them to be pursued by passers-by after their act of crime.
The "Little Jaffna" neighbourhood is a bustling district lined with shops and restaurants run by Sri Lankans, and Tamil is the main language heard on the streets.