AINISSA, Syria — At the headquarters of the Syrian Democratic Forces in Ainissa, the normally bustling offices were thinly populated on a recent day as the Kurdish-dominated coalition shifted its fighters to battle against the Turkish invasion in northwestern Syria.

Haqi Kobani, the deputy commander of the S.D.F., was holding down the fort in his capacious office, where a portrait of Abdullah Ocalan, the imprisoned leader of the Kurdistan Workers’ Party, or P.K.K., had pride of place. While Kurds hold most of the leadership positions, the S.D.F. is a majority Arab force now, engaged with the Americans in the fight against the Islamic State, mostly in Arab rather than Kurdish areas of Syria.

“Arabs love Abdullah Ocalan too,” Mr. Kobani said, casting a glance at the portrait. “There’s nothing hidden from our side. Everything we do is obvious and clear to the world.”

Many Arabs would probably differ about their love of Mr. Ocalan, whose socialist, radically egalitarian philosophy of governance holds sway throughout the autonomous region, known as Rojava, that the Kurds have carved out in Syria, with the help of the American-led international coalition. Their uneasy alliance, held together by the fight against the Islamic State, could be severely tested as the Kurds expand their control.