If a change is introduced to the timeline, then the chain that followed after the point the change occurred at will be discarded into this place and replaced with a new one that reflects the changes and how events would be influenced. This is confirmed to have been the fate of the Future, once Lavos was killed in 1999 AD.
The area called "Beyond Time's Eclipse" is presented in this game as a blue palette of the background in the Lavos Core fight.
After the party conquers the three Dimensional Vortex areas, they return to the End of Time, where Gaspar reveals that the bucket to the Day of Lavos recently received a second connection leading beyond Time's Eclipse, while still connecting to the Day of Lavos itself so that the player can still go there if they choose.
If the party chooses to follow this connection, they will find themselves at Time's Eclipse, represented by the background used when fighting Lavos' first form as well as when the Mammon Machine was fought in the Black Omen. There, an alternate version of Magus is encountered. After a monologue about there being a world for every potentiality, he opens a Gate that leads to this place, where the Dream Devourer easily defeats him in spite of his perseverance.
The Devourer also manages to shrug off the party's own attempt to defeat it, and is about to finish the job when the other Magus intervenes, causing Schala to awaken and send all of them away. The other Magus, in despair, erases his memories after this, potentially becoming Guile, though it's not clarified.
Upon arriving in the new Future after Lavos was killed, Belthasar founded Chronopolis and somehow discovered that Schala was imprisoned here. He formed Project Kid in order to try and get her out, and set up the Time Crash in order to accomplish this. Once everything was ready, he took the "Neo Epoch" to El Nido and watched his plan unfold as the events of the game.
Part of all this seemed to include sending Kid back in time to prevent Serge's death at Lynx's hands in 1010 AD, thereby causing the split in dimensions, as well as enlightening Serge and his friends to the existence of the Time Devourer once they reached Terra Tower.
Serge and his friends, after defeating the Dragon God atop Terra Tower, then head for Opassa Beach to use Kid's incomplete Time Egg (courtesy of the late Lucca before she died) and confront the Devourer directly. Of course, whether or not Belthasar's plan succeeds depends on whether or not the player created the Chrono Cross prior to going to Opassa Beach.
The true ending of the game sees the Chrono Cross created, successfully set up, and used during the battle with the Devourer, which destroys Lavos and frees Schala.
- It's not explicitly stated what the relationship is between the Dream Devourer of Chrono Trigger and the Time Devourer of Chrono Cross. Both are amalgamations of Lavos and Schala, but the Dream Devourer more closely resembles Lavos. This could be a design difference, as the Dream Devourer is based on Lavos' Chrono Trigger design, and the Time Devourer was designed by a separate party.
- The various names for this place could be the result of translation differences, especially between Chrono Cross and Chrono Trigger DS.