Seeing as how PJI are badly understood, I'll echo the "enemy has something to do with phase space/jumping" crew. I wouldn't be surprised if the PJIs are a sort of portal to phase space, which I imagine could possibly be a sort of parallel dimension of sorts (I need to read up on Sins lore more, apologizes if this is way off), containing their own beings, which would explain the horrified survivors of the Vasari fleet.
My guess at this point as well is that the creators of the PJI are the chasers. Perhaps some race that long ago concealed themselves in phase space for protection from war. Perhaps at the time, they were the only race in the galaxy to have mastered phase travel and so believed that permanent residence there would provide safety from an endless war they were fighting. Thousands of years later, the Vasari Empire rises and discovers these strange objects floating throughout the cosmos that impede the passage of ships into phase space.
One can imagine then that for a race that is attempting to evacuate realspace, PJI's would be quite important. Of course, as the VR have recently discovered, it doesn't just slow things in nearby realspace but can also block anything from entering nearby phasespace.
We can presume that there's some long-term issues with being in phase space. At minimum, we know that phase space will drop a ship's antimatter storage though we're not sure why (though the Advent appear to understand to a limited extent and the VR can fix this issue entirely). On top of that, after living in phase space for so long, perhaps they've forgotten how to fight in realspace and can only fight by yanking enemies into phase space. This sort of technology would certainly terrify the Vasari as they would just discover how far from mastering phase space they really were, not to mention the ability of an enemy to warp between places in realspace at will would itself be terrifying.
In-grav phase jumping is already something that can be easily seen to be a very great tactical advantage. Just look at Subverters and the Vorastra. The ability to teleport just a few ships at will is very powerful. Imagine an entire race based around that. An entire race that can rip things out of phase space at will. Phase Out Hull already damages shields and that's coming from the Vasari. Image what the PJI Builders' tech could do. Shields would probably get obliterated, leaving only hull and armor to protect the ship. Heck, the Antorak may have even been built to mirror the technology seen by survivors.
Presented with such an advanced enemy, the Vasari would have no choice to run as fast as they can. Kosturas previously used to bring rebellious sectors back under control would instead be used to catapult the Exodus Fleet as far as possible. To try to run in realspace would be hopeless as it is clearly faster to travel through phase space. Thus, the Vasari would have to be constantly afraid of reentering realspace because they'd be giving their phase space bound opponent a chance to catch up.
This may also explain why the Vasari are so concerned with running at this point and they way they intend to do it. The VL just want to run as fast as possible: cut down on the lag between getting out of phase space and returning to its safety. The VR on the other hand perhaps are more interested in being able to make a stand if it comes to it. They still want to run but perhaps they are afraid that with the chasers nearing, they may have to make a stand. This would explain then why they are wishing to retain AM reserves in phase space: they don't want to be caught defenseless without their ships' abilities.
In addition, it appears that gravity has some role in phase travel as creating a warp field is known to overcome Phase Jump Inhibitors' passive disruptive aura. This would also explain Gravity Pulse. Perhaps it is designed to either knock enemy ships out of phase space or to damage them while they're in there. The description in the manual after all says that at full power, this is capable of tearing other ships apart.
