I agree with what John Cross said here - multiple domains means more work. If there is a business case to justify that increase in work, then that is an easier decision. If there isn't enough business case to justify the work, then maybe from an SEO standpoint you should keep it on the same domain to get the new content ranking more quickly.
Along with SEO considerations, though, there are a few other ways to break down this question...
First, what are the user expectations? Yes, the products are different and not highly related but are the customers different? In the Tesco example, would people who are interested in groceries also be interested in banking? Or, put another way, would people who are interested in groceries (but not in banking) be offended to see that this company also offers banking services? If the users are interconnected or are (at minimum) not put off by the variety of products, then why not have everything on one domain? That way you get the strong SEO benefit of using sub-directories. This isn't always a cheap investment though, as it requires a strong architecture to keep the directories and content types/voices distinct, but totally doable and a good solution from an SEO standpoint.
Second, I'd look at this from a brand perspective. Is this all the same company delivering these goods? Is it all Tesco or Sainsburys? If it is the same brand name, then why not have everything live on one authoritative domain name (assuming you aren't going to chase away customers by showing the breadth of products offered)? Google is an example of this - look at the wide variety of services they offer mail, analytics, drive, G+, search, etc. - it is all Google, even though they offer a wide range of products to a diverse range of customers. Now, if New Product A is a different brand and a really different thing from anything else being done by the company (in Google's case - Android), then that likely justifies a separate domain and a larger business investment (not just for SEO, but for design and other types of marketing too).
Finally, you do need to look at this technically I think. Chances are that Tesco Bank has to live on a different domain just because of security considerations. Some times the technology limitations have to dictate what we do with SEO. If those are great enough, then we may have to do the work to create two distinct domains and get those domains earning rankings/traffic. In that case, the business/technical needs justify the work required.
Hope that helps!