To progress further into the Easter Egg, all four players will need to acquire and upgrade Apothicon Swords.
Start by taking the train from the Waterfront District to the Footlight District. If you are facing the direction of which the train is going, go to the right windows and look out them the ENTIRE train ride. You will see two glimpses of cuneiform code glowing red inside the building windows.
These can easily be missed so be sure to jot them down (as the images change per game, but not the locations). There are a total of two on this ride.
After you found those, take the train from the Footlight District to the Canals District, following the same procedure. This time, only one cuneiform symbol will be seen. You should have all three symbols written down or memorized.
(Note: If you cannot find the symbols, consult this map which shows where roughly the symbols should be.)
Here are the possible symbols.
Go down into the subway and activate Beast Mode. To your right will be a brick wall displaying several red cuneiform symbols. Zap the three symbols in any order to reveal the sword holsters.
In human form, take the egg from the sword holsters. You must now charge the egg four times in four different locations by killing zombies. These four locations are Cthulhu statues hidden inside boxes that must be broken in Beast Mode. One is in the subway directly across from the egg you just grabbed, one is in Canals near the back, one is in Footlight on the ground underneath the perk machine, and the one in Waterfront is in the room just past the perk machine. The egg must be placed on each statue and have zombies killed near it until it starts glowing.
Once the egg has been charged fully, take the egg back to the sword holster in the subway and place it. Obtain your sword.
Now you must upgrade your sword. Take note of which character you are and which ritual altar your character belongs to. Nero belongs at Easy Street, Jack Vincent belongs to Canals District, Jessica belongs to Footlight District, and Floyd belongs to Waterfront District. Return to your character's ritual altar and accept the Arch-Ovum a ghost Keeper is offering.
There are four glittering red circles around the map that you can now interact with to charge your Arch-Ovum. One is just outside the gate at spawn, one is just outside the Ruby Rabbit at Canals District, one is just outside the Burlesque in Footlight District, and one is just outside the Boxing Gym at Waterfront District
Simply place the egg on a red circle and kill the Margwas that spawn. You must do this at all four locations, and it can only be done once per round. Multiple players can do this in one round, but not at the same time.
Once each spot is done, go back to your character's altar and the spectral Keeper will finally give you the upgraded sword.