Citizen Sleeper

Ending tied to the following conditions:
1. Complete preparations of the Sidreal without helping Lem work by watching Mina.
2. Follow Sidreal/Castor storyline and board the Sidreal with 1 ticket.

As far as I call tell both SOLO TICKET and THE THREE BODY PROBLEM are the only mutually exclusive ending achievements. All six other ending achievements (as well as all non-secret achievements) can be obtained in one playthrough without cloud or savegame tricks.

However these can be knocked out in an hour or so if these are your last two.
Spoilers below.

For SOLO TICKET: work on the Sidereal exclusively, do not babysit Mina, scout the Lowend to meet Castor, and when the Sidereal moves to the hub, complete Castor’s quest, and leave on the Sidereal by yourself, leaving Lem and Mina behind on the Eye.

For THE THREE BODY PROBLEM: Work on the Sidereal and babysit Mina fully before it is finished. Scout the Lowend to meet Castor, and when the Sidereal moves to the hub, get the tickets from Castor but then “bide for time” when it comes to telling Lem about the IDs. Do this a few times and eventually Sleeper will discard the IDs. Watch the Sidereal leave with all three of you remaining on the Eye.

Short version: Build the Sidereal ship, DON'T babysit Mina, get a Sidereal ID for yourself, and board the Sidereal.

Long version:

This is 1 of 4 endings/achievements (Solo Ticket, Left Behind, Long Journey, and Three Body) related to Lem & Mina and the Sidereal colony ship. You can't get all 4 endings/achievements in a single playthrough unless you mess with cloud saves/save files; I believe there are minimum 3 playthroughs required to get all 4 endings. Notes on this in the comments.

Go to the Shipyard and complete the Yard Hand wheel through either "Assist a Shipbuilder" or "Haul Materials". You'll then get a story bit where you're introduced to Lem & Mina. Choices here don't matter.

Go to Unit Assembly and complete the Junior Tech wheel by successfully completing "Systems Fitting" enough times, or complete the Welder's Mate wheel by successfully completing "Welder's Mate" enough times. (If you get a negative outcome 3 times, you'll fill the Quality control wheel and have to do the Shipyard portion again.) You'll get a story bit with Lem & Mina where you have to watch Mina while Lem works; again, dialogue choices don't matter. (This does not count towards "babysitting" Mina; babysitting is done specifically at their apartment.)

Lem & Mina's Unit will unlock in Lowend, with babysitting Mina being an option. DO NOT GO HERE.

Go to Sidereal Bridge and complete Interface action "Sidereal Fit-Up" enough times to fill the "Sidereal Horizon" wheel (24 sections total; positive outcomes count for 2 sections). After this is done, go through another dialogue with Lem (choices don't matter). A 3-cycle countdown "Sidereal Draw" will start on Shipyard, and a 6-cycle countdown "Moving the Horizon" will start at Sidereal Horizon. Keep yourself alive and healthy until the 3-cycle countdown finishes.

Go to the Lem & Mina icon at the bottom of the Eye to go through a story bit where - surprise - you weren't even eligible to win the tickets. Continue to wait out the 6-cycle “Moving the Horizon” countdown.

After that, take the Ascender car up to the central hub, where there's a new icon for Sidereal Dock.

Warning: the next step will start an 8-cycle countdown timer; by the end of this timer, you will need to complete the section after that. Not hard, but just make sure you're not also chasing a bunch of other quests at the same time.

Repeat Intuit action "Scout the Dock" enough times to fill the 6-section "A Way On Board" wheel. After completing this step, you'll have a dialogue with Castor, who makes you an offer to get you onto the Sidereal. This starts an 8-cycle countdown; you will need to complete the section below before this timer runs out.

Go to Celis Ship at the bottom of the Eye. Repeat Engage action "Bluff Past Security" and/or Intuit action "Observe and Explore" enough times to fill the "Celis Infiltration“ wheel. Avoid negative outcomes as they will fill the negative "Celis Lockdown" wheel. After this is complete, go to Celis Servers and do critical action "Extract Celis Data". (A negative outcome still results in getting the data, but you'll lose 2 points of condition.)

Return to Castor, where you'll give him the data and he'll give you a Sidereal ID. Keep yourself alive and healthy until the aforementioned 8-cycle Sidereal Departure countdown finishes. After it does, go back up the Ascender Car to find Lem & Mina. Go through the story until you are given the option to “Call out.” or “Board.” For this achievement, choose “Board.” After the rest of the story, the achievement will unlock.

This ending kicks you to the main menu. Your autosave will be whatever state you were in before going up to the Sidereal Dock.