I had come to the same conclusion as Volt_Cruelerz since I read the lore of the Rebellion factions.
The TEC Rebels basically hate all aliens and want them dead, this makes alliances with either Advent or Vasari impossible, and alliances with TEC Loyalists unlikely as the loyalists would rather sit behind their walls in peace and would not cooperate to offensive campaigns on the Rebel's part.
The TEC Loyalists, on their part, want the war to end. Right now their strategy is to bunker down and wait until it blows over. If they were to be offered an alliance with the alien races, they would be inclined to accept as it would end hostilities.
The Vasari Loyalists, while they may be likely to accept alliances, it is in nobody's interest to ally with them as they intend to strip entire star systems clear of any resources and then GTFO. 50 / 50 chance that they would turn on their allies to steal their resources too, or just leave when things get too hot and you need their support.
The Vasari Rebels however, believe that their salvation lies in the other species and are actively seeking to form alliances with all parties. Obviously alliances with the VL would not work as they'd strip resources from them and their allies and then leave. TR would not be a good candidate either since they want all aliens dead.
The Advent Loyalists are still hellbent on the destruction of the TEC at any cost, and the conversion of all factions into the Unity. A temporary alliance with the Vasari may be possible in order to concentrate and exterminate the TEC but such an alliance would be risky and short lived, since when the TEC are gone the AL would most likely turn on the Vasari to try to force them into the Unity.
The Advent Rebels want to ensure the restoration and survival of the Unity. Basically, they want to secure the Unity's place in a balanced galaxy. Forcing others into the Unity would cause corruption, so there will be very few forceful conversion from them. Going to war with others would be controversial to the goal of preservation of the Unity since their defeat would mean the end of the Unity. Based on their defensive tech, it is safe to guess that they would be contempt with settling down in their corner of the galaxy and share it with the other two rather than try to take it all and risk being wiped out or corrupting the Unity in the process. With the abandonment of the desire for revenge, alliances with the TEC Loyalists, who also seek peace, are highly probably. Once these two have allied, accepting the Vasari Rebels' offer for cooperation would be logical, as it would bolster the strength of their alliance against the other factions and increases the chances for galactic peace.
So basically, the TEC loyalists, the the Advent Rebels and Vasari Rebels would form an alliance to finish off the other threee factions.
The TEC Rebels, Advent Loyalists and Vasari Loyalists would be unable to form their own alliance since they seek destruction of one another. The Trinity alliance would destroy them one at a time, since their combined forces would easily overcome any of the three's lone forces.
Assuming this happens and they then survive the coming of whatever it is that's chasing the Vasari, they may not remain as an alliance since they would no longer have a common enemy, but I believe that they would at least have peace. None of them desire war. The Advent would continue to develop their culture and technology and accept willing disciples of other factions who now wish to join the Unity of their own accord. The Vasari would finally get to settle down and begin to rebuild their race, 10 000 years of exodus can leave you in quite a sorry state. The TEC would become once again the economic empire it once was, allowing trade with Advent technology, now become available to them and provide aid for the Vasari to rebuild their civilization.
....then the Reapers show up and find out, as they run into star bases and Titans and massive fleets of ships larger than themselves, that they are horribly, monstrously outgunned.