Citizen Sleeper

This appears to be the only ending that cannot be achieved in the same playthrough as the other endings.

Follow the lengthy Lem/Mina/Sidreal/Castor missions until you have stolen the Celis data from the ship at the shipyard and then have an active scene with Lem and Mina. DO NOT tell Lem about the ID's 3 times and Sleeper will crumple and discard them at the end of the scene. Next, simply wait until the Sidreal departure date and meet Lem and Mina at the departure gate for this ending.

As far as I call tell both THE THREE BODY PROBLEM and SOLO TICKET 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 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 station.

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 without Lem and Mina.

Short version: Build the Sidereal ship, babysit Mina, get Sidereal IDs for everyone, but destroy the IDs.

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. Complete action "Babysit Mina" enough times to fill the "A Ticket Out" wheel. Go through another dialogue with Lem where choices don't matter. Make sure to complete all of Mina's babysitting before completing the "Sidereal Horizon" wheel, which is detailed next.

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. Afterwards, there will be a 3-cycle countdown timer on Lem & Mina's unit. Wait it out.

Talk to Lem at the apartment, choices don't matter. Then 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.)

Warning: dialogue choices DO MATTER in the dialogue with Lem detailed below.

Return to Castor, where you'll give him the data and he'll give you the IDs. This will follow immediately with a conversation with Lem. The first option between Sit and Stand doesn't matter; after that, choose these options when they appear:
Play for Time.
"Nothing. Forget it."
"I'm good. Just tired."
Destroy the IDs.

After this dialogue ends, keep yourself alive and healthy while the countdown timer runs out at the Sidereal Dock. After it does, go back up the Ascender Car to find Lem & Mina. After the story bit where the three of you watch the Sidereal depart the Eye, the achievement will unlock.