Note: this is the full text transcript of an image post on Imgur and Reddit. Links to the full post are on my profile if you're interested in checking it out!
Disclaimer: though I've tried to adhere to the canon of the game as much as possible, this is purely my idea of how such an ending would be presented in-game.
The prerequisites to beginning the run would involve completing a specific set of events in the run-up to the 'True Pacifist' ending. One of these would be finding a way to restore the Core to its original layout, the way it was before Mettaton reshuffled it. The purpose of these events is to restore the Underground to a unique state of equilibrium where 'he' is able to briefly emerge and take a partially physical form.
The run is activated by speaking to Asriel in the Ruins after your battle with him in the 'True Pacifist' run. After delivering his usual dialogue, if the prerequisites are completed, he will continue the dialogue with this:
Asriel:
* But first, can I just ask one last thing of you?
* Will you make sure you keep our meeting a secret from Mom and Dad?
* It would be easier if they thought I had never returned like this.
* I'd really appreciate it.
* …
* Well… you'd better get going.
Text prompt:
* Leave Asriel?
Leave
Asriel:
* Thank you.
* I'm so happy I was able to meet such a trustworthy person.
* Take care of Mom and Dad for me, ok?
Choosing this option reverts the game to a regular True Pacifist ending.
Do not
Asriel:
* Frisk... I...
* ...
* ... I think I understand how you feel.
* It's not an easy thing to ask of you, I know.
* But, please understand...
* I did some horrible things as a flower...
* And because I felt nothing, I was incapable of loving anyone.
* Not even my parents.
* But if they ever discovered I had come back, as the son they once knew...
* if only for a moment...
* it would hurt them, I think.
* ...
* I don't want to hurt anyone anymore.
* Not you, not them, no-one.
* That's why it needs to be this way...
* ...
* What's that expression for?
Speaking to Asriel again continues the conversation.
Asriel (2nd convo):
* You...
* You can see right through me, can't you Frisk?
* He...heh heh...
* ...
* You know...
* You remember I said that you were the sort of friend I wish I'd always had?
* I meant it.
* And the truth is...
* I really wish it didn't have to be this way.
* There are still so many things I want from this world.
* I want to see Mom and Dad again.
* I want to make amends for the pain I caused.
* But, right now, what I want most of all...
* ...is to be your friend.
* To stay here with you. Just like this.
* I-I want that more than anything!
* ...
* But these things can never happen.
* My soul was lost the day I returned from the surface.
* ...The day I died.
* Without a soul, I will go back to how I was.
* Nothing can change that now...
* …
* The flowers are beautiful today, aren't they?
Speaking to Asriel one more time will activate the run.
Asriel (3rd convo):
* You don't need to worry about me, Frisk.
* I'm fully prepared for what comes next.
* And it really was wonderful meeting you.
* Thank you for everything you've done.
* ...
* Goodbye.
Trying to speak to Asriel after this will give you the line (looping):
Asriel:
* Frisk...
* Don't you have anything better to do?