Relics of the Empire - Insomnia
First Available: Chapter 14
Other Requirements: FF15 Royal Edition DLC
Location: Insomnia Kingsglaive Base Camp
Rewards: EXP & The Tall's Talisman
This quest is part of the Final Fantasy 15 Royal Edition DLC; without that DLC this quest as well as much of the extra stuff in Insomnia will be unavailable. If you're confused about the very first part of this quest requiring you to speak to Josef, go up the broken escalator to find him; he's on the floor right above Cor.
Once the quest is updated you'll be tasked with destroying the three Imperial Bases which are scattered around Insomnia. You won't be able to just run inside of the base to destroy it, instead you'll have to go up into a guard tower and speak to Cor who will be there waiting for you.
At each of the three Imperial Bases you'll have to do something different to advance the story forward, sometimes it's just Noctis and Cor and other times you have your whole team.
Loot in Northeastern Imperial Base: Magnetron (found on the ground in the back near the fence)
Loot in Northwestern Imperial Base: Mythril Shaft (southwestern portion of the Imperial Base behind a fence)
Loot in Southeastern Imperial Base: Wizard Shield (found on the ground in the back of the base; in a similar location as the Magnetron) and Magitek Shield (found in the southeastern most corner of the base)
Completion of this quest as well as Crown City Without A King, and A Glaive Out of Time quests will reward you with an extremely powerful accessory for one of your bros. Make sure to go into your inventory and equip it immediately (yes, it's that good).