This story occurs in the universe of s/5782108/1/Harry_Potter_and_the_Methods_of_Rationality, written after reading Ch. 101. It contains spoilers for that story through that chapter, and is unlikely to make much sense without having read through at least Ch. 89 or so.
[passage from this site, s/5782108/82/Harry-Potter-and-the-Methods-of-Rationality]
...Phoenix travel was a sensation entirely unlike Apparition or portkeys. You caught on fire - you definitely felt yourself catching on fire, even though there was no pain - and instead of burning to ashes, the fire burned all the way through you and you became fire, and then you went out in one place and blazed up in another. It didn't sicken the stomach like portkeys or Apparition, but it was a rather unnerving experience nonetheless. If the underlying truth of phoenix travel really was becoming a specific instantiation of a more general Fire, then that seemed to hint you could potentially burn anywhere - even in the distant past, or in another universe, or in two places at once. You might go out in one place and blaze up in a hundred others, and the you who arrived at Hogwarts would never know the difference. Though Harry had read what he could about phoenixes, trying to figure out how to get one of his own, and there'd been no hint of anything remotely like that capability...
1:00AM, October 7th, 1992
Harry piloted the Zodiac through the frigid water, while small waves gently lifted and lowered the boat. There was near total darkness, he used only the stars to keep a consistent heading. He turned to the phoenix sitting next to him and whispered "now?" The phoenix shook its head.
Harry had woken up that morning, gone to meals and classes as if it were any other day. He didn't know what wizards used for calendars, and had actually forgotten how important that day was until that evening. He had had to use the final turn of his time-turner and could only go back to 11PM. But that had been sufficient. He had been awake since 7:00 that morning, and even with his adjusted sleep cycle, 24 hours awake was pushing it.
Two minutes passed. Harry turned and quietly asked "can we go now?" The phoenix nodded. Harry stood, the phoenix perched on his shoulder. He quietly said "Midnight, April 17th, 1992, Madam Pomfrey's Storeroom, after my next spell." The phoenix nodded again. Harry pointed his wand at the boat, mentally prepared himself for what he was about to see, and cast "Finite Incantatem". The boat reverted to the block of ice from which it had been transfigured as the boy and phoenix disappeared in a crack of fire.
Harry had not yet had much opportunity to experiment with phoenix travel. He had asked the phoenix to take him to two places simultaneously, unsuccessfully. His best interpretation was that the phoenix believed in some kind of conservation law whereby it had to rematerialize the same number and identity of people who had dematerialized, rather than anything related to matter or energy. Whether this was a physical law or the phoenix's conceptual limitation was unclear, but probably didn't matter, because trying to teach physics to a phoenix seemed like a poor use of time. Harry still believed in his previous intuition that phoenixes were capable of things that wizards were unaware of, and this one was more than willing to try. He decided that if he could go backwards in time more than 6 hours, that would be sufficient evidence for his hypothesis to try the rest of his plan.
Midnight, April 17th, 1992
They arrived successfully in the storeroom next to Hermione's still frozen corpse. The phoenix looked noticeably drained, but still near its peak magical capability. Harry pulled the 3-person broomstick from his pouch, commanded it up, moved Hermione's body into a set of straps he had previously designed.
When Harry approached Azkaban with Professor Quirrell, he had become very aware that the invisibility cloak did not have to cover every inch of something to disappear it. Later, Harry had done a few experiments to determine what the cloak would handle. It turned out that the cloak treated a collection of objects in a bag as a single thing if the invisibility cloak could wrap around part of the bag. Therefore, now he wrapped a large sheet around everything on the broomstick, including himself and the phoenix, and put on his invisibility cloak over that. Finally, he cast a quieting charm around the broom.
He then told the phoenix "12:07 PM, April 16th, 1992, Hogwarts roof whenever you're ready". The now-familiar crack of fire followed.
12:07 PM, April 16th, 1992
Harry hadn't had to do anything mentally demanding for the previous hour talk, so he'd been able to talk to the phoenix. He had explained the next phase of the plan three different ways and each time asked the phoenix if it understood, and if its understanding was consistent with the previous explanations. It was the best he could think of to prevent the illusion of transparency.
After arriving, Harry quickly flew the broom to what he remembered to be the correct roof access door. And then waited. And waited. This period of time had seemed like only an instant the first time around but must have taken 10-20 minutes. Or maybe this had just been a really slow minute, Harry couldn't read his watch. Finally, the door flew open and Hermione ran out. Harry forced himself to suppress his emotions and stick to the plan. He matched the speed of the broom to her pace, reached out with one hand, and touched her shoulder...
An outside observer would have noticed only a flicker, as Hermione Thomas Granger slowed down and began to relax, having reached the safety of sunlight, for Time could not be changed. But a significant change had occurred under the cloak, where a phoenix-powered "transporter accident" resulted in an identical Hermione William Granger safely aboard the U.S.S. Potemkin, which Harry had recently decided to name his three-person broom.
Harry angled the broom toward the edge of the roof, while calling "Hermione, can you hear me?"
"I'm right here Harry, stop yelling. And I know that I'm not good at broomstick flying but is this harness really necessary? How did you even get me into it?"
Harry pulled the broom alongside the castle just below roof level, so none of them would have to witness the scene above. "Sorry, adrenaline. Yes, it was. Magic, of course. How do you feel?"
"Well, my robes are wet and cold. And torn. That's weird. And my legs hurt from running. Actually, I'm pretty cold."
Harry pulled out his wand and waved it backwards, "thermos".
"Thanks. Also, thanks for the rescue attempt, but trolls can't pursue someone into sunlight, so it was really unnecessary... I don't have my wand, I must have dropped it on the roof, take us back up so I can get it."
"Not that troll, it was charmed against sunlight. And also, your broom, cloak, and mirror were sabotaged. So we'll have to wait until later to get your wand. Unrelatedly, let me know if you feel anything strange in the next few minutes." Harry had decided that just in case Dumbledore's belief in souls had any basis in fact, Hermione William Granger should be near Hermione Thomas Granger so she might absorb the imminent burst of magic.
"By the way, do you have enough magic to get us home?" Harry was concerned the phoenix would need to regenerate before they could leave and as he had already been awake for close to 25 hours, he was not looking forward to finding a place for them to hide and wait.
Fortunately, the phoenix cawed a 'yes', over Hermione saying "What do you mean, we're already at Hogw- was that a phoenix? What on earth is going on?"
Then they felt a burst of magic pass by. "That was strange. Were you expecting that?"
"Yes. What did you feel?" Harry hypothesized that if Hermione reported something qualitatively different than what he sensed, for example, the burst flowing into her and staying there, rather than just passing by, it would be at least weak evidence for souls.
"It felt like magic, and memories and emotions, and it felt familiar, and then it was gone. Do you know what it was?"
"Yes. I'm really tired now though, Let's go home."
11PM, October 6th, 1992
In a flash, it was night, and they were above the roof of Ravenclaw tower. "...Did we just travel through time? Phoenixes can do that?" While Hermione was talking and in the silence afterwards, Harry flew the broom through the hallways back to his dorm and opened his trunk.
After closing the trunk behind him, "Yeah, but that's a secret. No one else in the world knows that. Actually, I learned to do three allegedly impossible things today and put them together to accomplish a fourth. Which is stretching it, even for me. Although the phoenix really did most of the heavy lifting." The phoenix, understandably drained by the strenuous day, chose that moment to flame out into a pile of ash.
"Speaking of secrets, tomorrow I'm teaching you the Patronus Charm... after everything else you have to do tomorrow. It's long past time you learned. Right now, though, you should stay under the invisibility cloak so no one knows you're here. It's probably relatively safe now, but I don't want to deal with everything without eightish hours of sleep. I've been sleeping on a cot in my trunk for the past few months for an extra layer of security. I strongly recommend you stay in here too."
"This is Hogwarts, Harry. It's probably the safest place in the world. Serious injuries are rare, and no one's died here in decades."
"With one massive exception. Which will be the topic of every conversation you have for about the next week. Try to enjoy the ignorance, you'll feel nostalgic for it later. Anyway, there are plenty of books you can read until you get tired. Which will probably be around breakfast time. You're going to be jet-lagged for a while." Harry collapsed onto the cot. He pulled his pouch off an put it on the floor. Then he opened it, named each of his textbooks, and slid them toward her along the floor. "You should probably study. You're way behind in all your classes."
Harry was already asleep.