You need a ,refugee travel document. If you travel without it, they will you have forfeited your asylum status. They may not grant re-entry. Ask your immigration attorney.

REFUGEE TRAVEL DOCUMENT IT TAKES 15 to 18 months to process DO NOT LEAVE THE COUNTRY WITH OUT IT !!!!

If you have a valid passport i believe you can travel with it plus your green card. do not travel to your home country however. even with RTD. during naturalization travel to home country will pose trouble.