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They absolutely should, but what would be their impact on American companies buying each other out?

If they want to operate in the EU, quite a lot. See cases against Microsoft, Apple for example.

OK, can you provide specific cases/articles about it? I'm curious what can EU do with a merger of companies that are not legal identities within EU.

It's different for their local counterparts, but the original companies?

Microsoft/Activision Blizzard was tough for Microsoft in the EU despite them both being US companies, because they affected European consumers.