Earlier this summer the UK's second-largest supermarket delisted Princes tuna and Unilever brand I Can't Believe It's Not Butter in the start of a review of its product line-up.
Similar trials are also understood to have taken place in the soft drinks and water categories.
And while declining to specify where the next reviews were likely to take place, food trading director Darren Blackhurst promised "across ambient, certainly".
Speaking at a conference for Asda's 350 store managers, Blackhurst explained the trials for Princes tuna and I Can't Believe It's Not Butter had been successful.
"We are challenging whether certain brands have brand equity. We took Princes out. The tuna don't swim past and say 'bugger me, I don't want to go in that can, I want to go in this one'. So we chose John West. And have we had a single complaint from customers? Not one. Why? Because it wasn't a brand."
Blackhurst admitted that other trials had been a resounding validation of well-known brands.
"We tried it with McVitie's Homewheat. That's a brand," he said. "We're bringing that back. And we delisted Baxters Soup in Scotland. That's also back."
This was proof that the exercise was a genuine attempt to test which brands were valued by customers.
"This initiative is about bringing back real choice to the customer, with more local, premium and organic products. It's clear it works for customers and for us, who can bring in more volume so costs go down, and for suppliers too, who enjoy disproportionate growth."
Blackhurst said that the increased focus on local produce was also benefiting the environment, with an estimated 7m road miles saved already following the introduction of 15 hubs stocking nearly 5,000 lines from 400 suppliers.
Sales of local lines were, in some cases outselling major national brands. In Newcastle Doddington Dairy ice creams were oustripping sales of Häagen-Dazs, while local sausages around the new Kendal hub at Plumgarth were selling better than Wall's.
Indeed, with sales of Seabrook Crisps now outperforming national brand Walkers in the Yorkshire area, Blackhurst said it was possible that even the UK's No1 crisp brand would be challenged. "We have a message for Walkers. You're going to have to fight for the right. Don't ignore this.
"Where there's a local producer and it outclasses the major brand, in that shop we'll be clear who the leader is. Admittedly, it would take a lot of local crisp manufacturers to challenge a national market leader like Walkers - and personally I think it's a fantastic product - but I'm sensing in a lot of shops a change in attitude towards local suppliers and away from national brands. It's an interesting time.
"In a supermarket you need national brands and to be locally relevant. Being truly relevant isn't about national campaigns, it's about one individual in one store," Blackhurst added.
A day in the life of Andy Bond p30