For a 12th - 14th century (Crusades-era) Mamluk
Maille, lamellar armour, or armour of leather bands for torso defense.
Laminated leather helmet, simple conical helmets, turban helmet.
For a 17th century Mamluk cavalryman (or Ottoman sipahi)
Maille, mail-and-plate armour (yushman or krug), bazubands, thigh armour.
Conical helmet with ear-plates, sliding nasal, mail or lobster-tail aventail, or turban helmet.
There are a few existing garments which could be "eastern gambesons", but it is not so evident as in Western Europe. Something must be worn under armour to have a decent protection but the sources emphasize it less in the East. In my opinion we can reach the same effect with textile bandage (like the turban) on the limbs. E.g. instead of an arming coat, I use bandage under my basubands,