SHANNON, Ireland — When it came time to stretch his legs during a refueling stop in Ireland on his recent Europe trip, Vice President Mike Pence left his cabin suite at the front of Air Force Two dressed in a flight jacket with the vice-presidential seal.

As he disembarked to stroll a few laps around the airport terminal like a Midwestern mall walker, his wife, Karen, made a beeline to an assortment of souvenirs in the duty-free shop.

Back across the Atlantic, President Trump, Mr. Pence’s boss, was in the midst of firing off his early morning tweets, including one addressed to the Senate Intelligence Committee denying, once again, that there was any collusion between the Trump campaign and Russia. The vice president, for his part, seemed worlds away, enjoying a brief stop with the second lady as if they were tourists on a long-planned vacation.

Appearing together in Washington, Mr. Trump, a former reality show mogul, and Mr. Pence, an evangelical Christian, are a strikingly odd couple, with the vice president usually seen as a meek helpmate casting a cipher’s gaze in the president’s direction. But with at least 10 diplomatic trips to six continents under his belt, Mr. Pence appears to have mastered the art of stepping out of Mr. Trump’s shadow on his own terms. He avoids taking any perilous steps into the president’s limelight by staying relentlessly on message, whatever the diplomatic cost.