Three bodies have so far been recovered after the men from West Bengal's Sunderbans region defied bad weather alerts from local authorities and set off early Monday.
Rescue teams on hovercrafts and an aircraft were searching the area.
"We fear that all 19 fishermen are dead," coast guard official M. A. Warsi said.
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West Bengal chief minister Mamata Banerjee said her government would respond by regulating the fishing industry and enforcing the wearing of life jackets on boats.
The Sunderbans, a Unesco world heritage site famed for its mangrove forest is home to thousands of fishermen who fish throughout the year.
But the onset of the annual monsoon season sets off bad weather and storms in the coastal region, making sea voyages risky.
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