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The similarities to Gleichshaltung in the Germany of the late 30s greatly unsettle me

See now, that's where it just gets silly.

Read up on the history of the EU and the men who were the driving force behind its foundation.

You'll find that all of them were committed anti-fascists, some actively fought against the Nazis, some were imprisoned by the Nazis, others by Mussolini, others contributed greatly to the Allied war effort.

You'll also find that some of the key figures behind the formation of the EU were deeply committed Catholics who were greatly influenced by late nineteenth- and early twentieth-century Catholic social teaching.

The EU owes a lot more to Rerum Novarum and Quadragesimo Anno than it does to Gleichshaltung.