Here's my best guess that will probably be so far off of Valve's story, but speculating is fun non the less.
Warning, below contains spoilers from Half-Life: Episode 2 and previous games, along the Portal duology.
Leaving off from Eli kicking the bucket, Alyx and Gordon make it a new goal to eradicate the advisers, out of vengeance. However, their primary goal remains to locate Dr. Mossman and the Borealis. Upon finding and reaching the Borealis the player is faced with a decision. Eli's dying wish was for Gordon to ensure that the Borealis is destroyed, whatever the cost. Destroying the Borealis will result in a personal loss to Gordon (Presumably not Alyx as killing such an important character would not go over well unless it was the intended goal all along that the player would chose her).
The destruction of the Borealis would ensure that the Combine could not utilize Aperture technology to create a new portal. Perhaps, aboard the Borealis, Alyx and Gordon discover the coordinates of the Aperture facility. They intend to destroy everything to do with Apature to make sure that the technology can not fall to the wrong hands. Alyx and Gordon find themselves in a field with a small shack. Alyx is frustrated as this is where the coordinates lead.
What I would love to say is that the find Chell, however, unless there is some drastic change in ideals of Valve games, the two protagonists will never cross paths as this would force Valve to give Chell a voice (the reason being that it's speculated the reason Chell does not speak is because she does not want to give GLaDOS the satisfaction of knowing she's getting to Chell) so it's likely if they were to ever enter Aperture main facility that they wouldn't run into Chell.
That's as far as I can go on speculation, please feel free to tear apart of fun story with your facts.
Side note, my greatest wish for Half Life 3 is that they make the decision to release it without advertising, bear with me. It would be the most fascinating thing to watch google map point data and see when the first person finds it. All they need to do is simply place it on steam. Don't mention it, don't advertise it, nothing. Just put it there, normal price and everything just wait for the first one to buy it. It would be amazing.
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