WASHINGTON — As CEO of Exxon Mobil, Rex Tillerson was accustomed to giving orders and brokering multibillion-dollar deals around the globe.

Now he leads a State Department that must handle the fallout from President Donald Trump’s disruptive phone calls with leaders in Mexico and Australia, provocative comments about NATO and China, and the subsequently blocked order suspending travel from seven Muslim-majority nations.

Secretary of State Tillerson made his debut on the global stage Thursday at a G-20 summit in Bonn, Germany, saying little in public but working behind the scenes to reassure a dozen or so foreign ministers that U.S. foreign policy was nothing to fear.

Tillerson faced a barrage of questions from his foreign counterparts on whether Trump administration policy would hew to traditional lines or follow Trump’s sometimes ad hoc pronouncements. The diplomat mostly stuck to a handful of carefully crafted policy statements.

He described his first formal meeting with Russian Foreign Minister Sergey Lavrov as “productive” and avoided any mention of the FBI investigation back home into whether Trump’s campaign team or associates had improper contacts with Russian officials.

Tillerson apparently has the president’s ear.

After he spoke to Trump, the White House issued a mild rebuke for the first time to Israel for expanding settlements in the disputed West Bank. Trump repeated the criticism Wednesday at a joint news conference with visiting Israeli Prime Minister Benjamin Netanyahu.

And Tillerson is winning over skeptics at the State Department who questioned his lack of government experience.

Several officials who have briefed Tillerson say he asks questions and listens patiently. Predecessor John Kerry, a former U.S. senator with vast diplomatic experience, tended to pontificate rather than listen, the officials said.

Still, with Trump’s foreign policy a work in progress, Tillerson has kept a low public profile since he was sworn in Feb. 1.

America’s top diplomat was mum when the Trump administration slapped new sanctions on Iran for launching a ballistic missile. He also was silent when North Korea launched a mid-range ballistic missile last week, letting the White House respond instead.

He has said nothing in public about renewed violence by Russian-backed separatists in eastern Ukraine, or the sighting of a Russian spy ship in international waters off the coast of New England.

One reason, perhaps, is Tillerson still doesn’t have the full team of advisers and aides who normally report to the secretary of state — including a deputy secretary, the official who runs day-to-day operations.

Critics worry about Tillerson’s friendship with Russian President Vladimir Putin, an issue that dominated his confirmation hearing, and whether he will convince Trump that the Russian frequently acts against U.S. interests.

It is not unusual for the White House to control foreign policy either through the National Security Council or a small group of advisers and to marginalize the State Department.

Sharon Burke, a former State and Defense official in the Barack Obama and George W. Bush administrations, suggested that Tillerson has room to maneuver as a diplomat because Trump’s relatively mild actions overseas so far don’t match his rhetoric.

“They want (policies) to look like a sharp break from the past,” even if they aren’t, said Burke, now a senior fellow at the New America Foundation think tank.

White House Press Secretary Sean Spicer said Trump’s shifting foreign policy is deliberate, meant to keep his adversaries off guard. Trump “doesn’t like to telegraph his options,” Spicer told reporters.

tracy.wilkinson@latimes.com