WASHINGTON—President Trump’s response to the violent turns at some racial-justice protests in U.S. cities earlier this month has ushered in the most acute test of civil-military relations in decades.

Mr. Trump’s consideration of deploying active-duty troops to impose order—though ultimately not acted upon—drew objections from Defense Secretary Mark Esper and Army Gen. Mark Milley, the chairman of the Joint Chiefs of Staff.

And…

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