The Class 4 Administrative Zones are battle arenas where you will face large waves of enemies, then receive a reward at the end should you win. There are 10 of these zones in the game, and are unlocked by speaking to certain individuals in harbors or ports. The Van Eltia can sail to the location once it's been made available.
Class 4 Administrative Zone 1 - Aberrant Quarter
You can sail here after talking to a pirate once the party returns to Raneed from Lothringen Tower. Once you reach the location, be sure to look at the northeast corner for a Katz box worth 115 spirits. Participate in the battle and win to receive the Virtuoso's Ventite.
Class 4 Administrative Zone 2 - Beast Quarter
This one is given by a dude in Yseult's harbor after Haria Village is overtaken by malevolence. Once again, don't forget to check the area around the arena to find one more Katz box. This one costs 135 spirits to open. Compete in the arena and win for some Featherlight Steps.
Class 4 Administrative Zone 3 - Crustacean Quarter
This is unlocked by speaking with the pirate at the front docks of Titania Island Prison, after the party claims the place as their hideout. Check the area around the arena for a Katz box worth 150 spirits. Fight daemons on the arena and win to receive the Avenger's Ventite.
Class 4 Administrative Zone 4 - Winged Quarter
After returning to Titania from the Faldies Ruins, go through all the events until the group is ready to sail again. Go to the docks and check one of the two sub-event bubbles there to unlock this next zone. Look at the northern section of the area to find a 200 spirits Katz box. Compete and win at the arena to get a Force Ring
Class 4 Administrative Zone 5 - Apodous Quarter
After returning to Taliesin from Aball, head to the docks and speak to the pirate there. At the arena, check the northern section once again for the next Katz box worth 225 spirits this time. Winning the arena battle rewards you with a Military Hike.
Class 4 Administrative Zone 6 - Demihuman Quarter
I lost track on when exactly this arena can be unlocked but my estimate would be at some point after the events in Baird Marsh (first time), so try checking every harbor in the game around that time and see. At the arena, check the western section find a 230 spirits Katz box. Fight and win at the arena to receive a Blood Blade.
Class 4 Administrative Zone 7 - Undead Quarter
This one is unlocked by talking to the pirates in Meirchio just before you head for Mt. Killaraus. There, check the southwest section of the area to find a Katz box worth 265 spirits. Complete the arena battle for a Barrier Ring.
Class 4 Administrative Zone 8 - Fiend Quarter
This one is unlocked by talking to the pirates in Meirchio just before you head for Mt. Killaraus. There, check the west section of the area to find a Katz box worth 265 spirits. Complete the arena battle for a Mythril Waistcoat.
Class 4 Administrative Zone 9 - Armored Quarter
After completing the events at Mt. Killaraus, sail to Hellawes and speak to the pirate at the docks. Check the north end of the area for a Katz box that costs 375 spirits. The reward for completing the arena battle is a Haggler's Acerite.
Class 4 Administrative Zone 10 - Elite Quarter
After beating the game, warp to Port Zekson and speak to the pirate standing next to the Katz that lets you play mini-games. Now for the Katz box, simply look behind you to find it. This one opens in exchange for 450 spirits. Complete the arena battle for Adamantine Boots.