Aldi has its own brand name markets in the USA, not everywhere, but they seem to be common in the East and the South. They are much better known in the USA for their Trader Joe's market chain, which sells high quality, sort of upscale and somewhat exotic foods and wines. Aldi didn't invent Trader Joe's, it began in Pasadena CA in the 1960s, but Aldi acquired the stores (1990s). I think Trader Joe's does sell a DW soap, I ought to check next time there and see where their product is made. Does the Aldi detergent you buy in UK show a country of origin on the package?

http://www.traderjoes.com/

Tesco began a chain of markets (mainly in California to start) called "Fresh and Easy", which bear some similarity to Trader Joe's. Their forte seems to be prepared meals that one can take to the office and reheat. I've never been in one of the stores, but they have their fans. When the first ones opened here several years ago, British ex-pats flocked to them in the expectation that they would find "imported goods" (= biscuits, tea, clotted cream, bangers, Quality Street, etc.) at discount prices. Alas, the stores do not carry any "British goods" and the initial news coverage focused on the disappointment of the would-be ex-pat customers.

http://www.freshandeasy.com/