Harry can't escape the attention he gets in the wizarding world, and hasn't had time to learn to cope since the war ended. So, he went to then one place wizarding press are not legally allowed to go: the Muggle world. Now living comfortably hidden in the heart of Auckland, New Zealand, a city with no magical community to speak of, Harry takes comfort in writing letters to friends, spoiling Teddy and the Weasley children on holidays, and working various low-pressure Muggle jobs to stay busy.
He is sufficiently surprised, to say the least, when Fleur Delacour turns up at his doorstep asking concerning questions about soul magic. Through her work with magical artifacts, an accident at work lead to a curse attaching itself to her soul, ripping her away from the present reality and into another parallel universe for longer and longer periods. Right now, it's only a minute or two, but every disappearance is longer than the last.
Considering that all soul magic knowledge was suppressed in an effort to prevent the spread of its usage, Harry is the only living example of how soul magic affects the person to which it is attached. Feeling obligated to help people he considers family, he offers his services as a test subject and research assistant.
The research turns up a single unappealing treatment option: Lock Fleur into the parallel universe, but she would be trapped there.
This would be Flowerpot, and as the research continued Fleur and Harry would grow closer and closer, until Harry decides to join her in the alternate reality so she isn't alone. A good epilogue from the story would be to reveal that Harry and Fleur have figured out how to communicate with their former reality, and the final chapter of the fic is their letters back in forth Ron/Hermione/Gabrielle talking about their new life and what has happened while they were gone.