That was already clear even during the game itself - you actually fight some Garlean Frumentari soldiers in Eastern Thanalan after the raid on the Waking Sands during the original 2.0 story (the player catches a unit of them sneaking around the Church of Saint Adama Landama, presumably they had been trailing the player's character after fleeing Vesper Bay for Camp Drybone).

Either way, it was clear the Frumentari were the 'spooks' of the Garlean military even long before the lore book was available based on that.