Even if the Registers of Deeds COULD keep foreigners out (and I don't think there's any law against letting foreigners in), you can't control folks like DataTrace and PropInfo (might be Black Knight?  I don't remember...just know that they're an on-line plant like DataTrace).

The only way to put an end to this is get a law passed, and my Libertarian heart says that getting the government involved is never a good idea.