Last fall, Patrick Maxwell, a 29-year-old Iraq war veteran now selling real estate in this bustling city, saw something in news footage of Islamic fighters in Iraq that he never saw as an infantry Marine there: the enemy. “We patrolled every day, got shot at, mortared, hit by I.E.D.s, one of my friends was killed,” said Mr. Maxwell, a former sergeant who deployed in 2006 to Anbar Province. “But I never saw the enemy, never fired a shot.” With the Islamic State, also known as ISIS or ISIL, hoisting its black flag above many Iraqi cities that United States troops...