There are currently three ways to get Illuminati Dark Gobcoat (which upgrades waist, neck, earring, wrist, and ring slots) and Illuminati Taut Gobtwine (which upgrades head, body, hands, legs, and feet slots).

1. Alexander - Midas (Savage): If no one in the party has already cleared, the group will get one Gobcoat for clearing A6S and one Gobtwine for clearing A7S. In a party with four or fewer people who've already cleared, you'll instead have a 50/50 shot at them. Additionally, Sabina in Idyllshire sells Gobcoats in exchange for four Midan Manifesto - Page 2 and Gobtwines in exchange for four Midan Manifesto - Page 3. You'll get one page for your first weekly clear of A6S and A7S respectively.

2. Void Ark & Weeping City of Mhach: Bertana in Idyllshire sells Gobcoats and Gobtwine in exchange for Mhachi Farthings and Mhachi Pennies, obtained from clearing Void Ark and Weeping City of Mhach respectively. You will get one Gobcoat or Gobtwine in exchange for one farthing and one penny. Note that although you can obtain unlimited farthings, you'll only receive a penny for your first weekly clear of Weeping City (this restriction will likely be lifted in 3.4).

3. The Hunt: Bertana also sells Gobcoats and Gobtwine in exchange for four or six Unstained Clan Mark Logs respectively. The logs can be purchased from Ardolain, in Foundation, for 500 Centurio Seals each.

Once you've obtained the Gobcoat or Gobtwine, you'll need speak to Seika, in Idyllshire, and select the option corresponding to Lore or Eikon gear, select the item you want to upgrade, and trade both it and the enhancement item to Seika to receive the upgraded version.