In a private room in China’s Great Hall of the People in November, Secretary of State Rex Tillerson sat with President Donald Trump and other U.S. officials as their hosts delivered plates of wilted Caesar salad.

Mr. Trump, in the midst of a five-country tour of Asia, grew concerned the untouched greens would offend the Chinese, according to people familiar with the matter. So he ordered Mr. Tillerson to start. “Rex,” he said, “eat the salad.”