More than three quarters of consumers who buy tinned tuna believe the UK's major supermarkets should not sell brands that harm other wildlife, according to a survey.
The poll for Greenpeace found 78% of those who buy the fish believe retailers should stop sourcing from companies who catch it in a way that also kills sharks, rays and turtles.
It also found 84% of Tesco tinned tuna consumers thought that the supermarket giant made the right decision to "clean up" its own brand in 2012 and 70% said they opposed a scenario in which Tesco or any other retailer reintroduced unsustainable tuna to their shelves.
Last week Tesco was caught up in a new row over sustainable tuna after it emerged that it was stocking a cut-price brand which campaigners claimed harmed other wildlife.
Greenpeace said that after pledging to make sure all its own-brand tuna was sustainably caught in 2011, Tesco had started to stock Oriental & Pacific tinned tuna which the green group claimed was "dirty" because it is caught in a way that can kill sharks, rays and turtles.
Tesco was ranked in a "must try harder" category of a tuna league table drawn up by Greenpeace over the action the company is taking to make sure tinned tuna is sustainably caught.
Greenpeace said more than 70,000 people had signed a petition to Tesco chief executive Philip Clarke since the publication of the league table, calling on him to remove Oriental & Pacific from shelves.
Greenpeace UK oceans campaigner Ariana Densham said: "Tesco aren't doing the right thing for its customers or for endangered marine life. Tesco has justified selling Oriental & Pacific saying they want to give consumers choice but this shows they are completely out of touch.
"Consumers are clear - they don't want the 'choice' to buy dodgy tuna, caught in a way that kills sharks, rays and turtles. The tins don't even have clear labelling, so people don't know what they're buying. That's not choice, that's deception.
"Tesco seems to be willing to say anything to get its customers off its back. They say that they were one of the first to move to using sustainable tuna. This is fiction. Tesco were dragging their feet and was one of the last supermarkets to commit and only finally gave in after mass public pressure."
A Tesco spokesman said: "Customers have a great choice of sustainable tuna at Tesco.
"We insisted that Oriental & Pacific make a commitment - like other branded suppliers - to work towards a more sustainable fishing policy. They have since made that commitment."
Greenpeace said that tuna in the Oriental & Pacific brand was "dirty" because it was caught in big nets known as purse seines, using "fish aggregation devices" or floating rafts which attracted not just tuna but other species that are then accidentally caught in the nets.
LDH, the company which owns the Oriental & Pacific (O&P) tuna brand, said it "totally refutes" any claim it was dirty, which implied there was something wrong with the product itself or that the skipjack tuna in it was unsustainably caught.
In a statement the company, which supplies items such as canned tomato products, fish, fruit and vegetables and dried pasta, said: "At least 85% of the tuna we sell is fished using the pole and line method; our O&P brand skipjack tuna is caught using the purse seine fishing method, which accounts for 63% of all tuna caught around the globe.
"Credible scientific research by the Regional Fisheries Management Organisations shows that stocks of skipjack tuna are healthy.
"All of our tuna suppliers are members of the International Seafood Sustainability Foundation and support its research-led initiatives for long-term conservation of tuna stocks, reducing by-catch and promoting eco-system health.
"We are fully committed to supporting our supply base in these ongoing efforts and their work to support the scientific community in order to achieve best practice."
YouGov surveyed 2,240 adults online between March 7-10.