The last week of May, I went to Scotland for vacation. Since my family and I needed to get some supplies as we traveled the lovely countryside, we happened across a large store called Tesco, which seemed to fit the bill when it came to some items we needed to purchase -- like extra luggage for all that shortbread to bring home to the States. Once inside the store, my mom wondered whether Tesco was somehow related to Wal-Mart (NYSE:WMT), because of the similarities.

As it turns out, U.K.-based Tesco does have a connection to Wal-Mart -- namely, it's been giving the world's largest retailer a major headache in the U.K. And it's a retailer we're bound to hear more about as time goes on.

With my curiosity piqued, I noted with interest an article in today's Wall Street Journal about Tesco's success in using its loyalty program to lure customers. Through its Clubcard, it has been able to interpret -- on what appears to be a very granular level -- customer data in order to offer up deals on what its shoppers want.

The article states that this loyalty program has helped Tesco snap up 31% of the grocery-market share in Britain, compared with just 16% for Wal-Mart's U.K. alter ego, Asda, a chain that Wal-Mart purchased in 2000 for nearly $11 billion. Wal-Mart may be a giant here in the States, but it's got its work cut out for it in some international regions. Meanwhile, it's clear that Tesco is a major force in Britain: Last August, Wal-Mart even urged the British government to examine Tesco's market dominance, which, as Fool contributor Rich Smith pointed out, is such a delicious irony.

Tesco's clearly giving Wal-Mart a run for its money in terms of getting a good foothold in the U.K. The Scottish Tesco store I visited had a similar merchandise mix to a Wal-Mart Supercenter here in the U.S., if maybe slightly more upscale. The store was neat, clean, and bright, had quite a variety of merchandise, and was fully packed with shoppers standing in long lines. Some extra digging around reveals that Tesco has several different types of store formats, though.

What makes all of this even more interesting is that Tesco has plans to open stores here in the U.S., starting on the West Coast in 2007. According to a CNN article earlier this year, Tesco said it plans a store specially focused on the U.S. market, supposedly a cross between a grocer and a convenience store. That would perhaps most closely mirror its smaller Tesco Express stores, as opposed to its supercenters on the opposite end of the spectrum.

If that's the case, it seems that here in the U.S. -- at this juncture, anyway -- Tesco will threaten traditional grocers and convenience stores more than it will Wal-Mart and its low-cost brethren, Target (NYSE:TGT) and Costco (NASDAQ:COST), although it's obvious that Tesco could threaten them all, depending on the format it decides to take with the stores, and whether the public responds.

Anybody who's interested in stocks of grocers, discounters, and the like ought to keep a close eye on what Tesco's up to. Given that Tesco seems like a savvy, aggressive retailer indeed, this impending British invasion could add up to a lot of pain for some of its American counterparts.

Wal-Mart is a Motley Fool Inside Value selection, and Costco has been recommended by Motley Fool Stock Advisor . Check out any of our investing newsletters free for 30 days.

Alyce Lomax does not own shares of any of the companies mentioned.