Metallourgeiones are structures whose function is to refine ore, usually Areios ore in the field to provide materials for a field army. It is also the lingua franca term for such a said Areios Refinery.
While the official name of such a building was not given in the Mundane Exploration of Cyan Sphere of Doom, the building was featured on later stages where a seemingly abandoned automated Mobile Construction Yard was present.
The ore refinery appears in the game as Refinery, which is common to all 3 factions. It is the main way of generating income (and the only source of income for Territorial Independence Army and Posterity of The Sentinels).
The Territorial Independence Army spends 400 credits to build a Refinery, the Posterity of the Sentinels 450, and Evil Baby Corporation 300. If taking into account worker cost, the TIA and the Posterity spends 350, while the Evil Baby Corporation spends 260, meaning the EBC has the advantage in building refineries.