Oh yes, because I definitely am trolling all of you. 

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.

You people are so overthinking this...the game files give you the answer to your question...

Thus far, the game files, the lore, and the intro movie point to only one plausible possibility of cooperation, and that is Advent and TL working under the Vasari to deal with this unknown threat...it ain't an alliance to wipe out "the other side" and it isn't a friendship by any means...

In general, everyone hates everyone...the VR tolerate the other races out of practicality...there is no TL/AR thing going on, and there is no "uber alliance" to deal with the other factions...also half the time people use the word "alliance" on this thread, they probably should have used the word "armistice"...

As an example, the TL are not going to "ally" with the AR...they might sign an armistice and move on to other things but they ain't going to be working together and they sure as hell don't like each other...

Except that we know that the VR make nice with the other two races.  They have access to the supply chains of the TEC and shield tech from the Advent.  You are right however that the alliance wouldn't try to wipe out the other factions.

The Vasari, TL, and AR have no need for war.  The VL don't care what happens to everyone else.  They just want to run.  As for the TL and AR, they have no need for continued conflict with each other.  The TEC are still probably disgusted by the cultural practices of the Advent.  In fact, the AR are probably worse than before because they are all about purifying the Unity and will sacrifice their own people at will.  The VR are still slavers as well.  It's not like people are going to like each other, but it would be an alliance, even if a fragile one.

Let's say that the chasers catch up and the three ultimately defeat it.  I really doubt that they would resume fighting as soon as the war ended.  They'd just want the death to stop and by that time, there would probably be countless times where members of each race would have made desperate attempts to save their allies.  At first, it would simply be out of practicality, but over time, a feeling of cohesion would begin to exist.  Sure, they still won't like the cultures of the others, but if they're the guys keeping you alive, they can't be that bad.  Over time, this would likely lead to relatively peaceful relations between the factions.  

I'm not saying that they'll be friends, but they will be allies.  An alliance doesn't mean a pact.  It just means that they're on the same side.

In addition to what Volt said,

The goal of the alliance is not directly to take out "the other side". They are already at war with the "other side".

The goal of the alliance is to have peace and to be ready when the Vasari's doom gets there.

However, the TR, AL and VL are getting in the way of that peace, they won't stop attacking, except the factions they had been attacking just tripled in strength via alliance. In order to have peace, the others MUST and WILL be eliminated, it is inevitable.

They will be forced to cooperate in order to reach their goals, or else nothing will change. The enemy of my enemy is my friend.

Now now, we haven't gotten to the 4th expansion yet...

It's like this...

SoaSE: ORIGINAL

 SoaSE: ENTRENCHMENT

SoaSE: DIPLOMACY

SoaSE: REBELLION

 Presumably the next title will be SoaSE: KUMBAYA

Pat, I still disagree with you.  First off, I really doubt that the VL would ever be an issue in the game as they'll probably escape and keep on running.*  As for the AL and TR, They're going to keep going at it for some time as they don't want peace; they want domination.  I really doubt that the alliance would attempt to wipe the others out.  They'd really like to welcome the others in if at all possible, but they realize this isn't feasible.  As vicious as the TR for instance are and their secession from the TEC, I think the TL would be much more fond of them than the AR.  It's just that the TR aren't being reasonable.

*What would be awesome would be if in Sins II the chasers arrived and as expansions were released, it would help the story unfold more and more until the final expansion could have a story component wherein the VL return to finish it once and for all, having decided that they can't run forever and the Advent and TEC were their best chances for survival.  I can just imagine a beautiful opening cinematic where the chasers are annihilating alliance forces, destroying the Eradica* and Kultorask.  Just as the Ankylon is about to be destroyed, a Kostura shell slams into the planet and the Vorastra jumps in, followed by the rest of the Exodus Fleet.  Screen turns to black and Sins of a Solar Empire II Redemption appears onscreen.

**Technically, the Eradica is killed by the Coronata in Rebellion as shown in the opening cinematic when the Coronata blasts the Eradica with Unity Mass and Unyielding Will engages.  Given that it's the AR however, them rebuilding a titan is much more feasible as part of the story than the other races which only rebuild theirs due to gameplay value.

True for the VL. Seeing the superior forces of an alliance, they would likely choose to leave with what resources they already have.

As for the AL joining the AR in peace, and the TR doing the same with the TL, it would be a difficult and risky decision. While at peace, both AL and TR groups would still harness hostilities against the other species which could easily create terrorist groups trying to wipe out the others, which could degenerate into civil war and start the whole thing over again.

I'm not saying the Trinity alliance would gladly wipe out the others, but if they do not cease their attacks, it will bring about constant disruption of the preparations for whatever it is that's coming after the Vasari and a strain of the resources alloted to this effect.

I'm not saying they would want to, they just dislike their race generally less.