This is ridiculously easy to set up with the frame data code. From what I can tell it only works with Squall, not Belay for some reason. My theory is more or less that, if Nana is interrupted on the frame she is meant to stick to Popo, the game won't process the unsticking like normal and cause constant stickiness (Edit: I've tested a little further and as long as Nana is stuck to Popo, if you hit only her it will cause this glitch, even during the middle of Squall). Also I feel "sticking" is a more appropriate name for this than "fusion," since Nana's just attached to Popo permanently, not actually fused with him, they're not permanently synched or anything.
I don't see TOO much of a playstyle change from this. Basically Nana is stuck to you so she can't get launched but she still has her delay and whatnot so normal desynching etc. doesn't change, also certain circumstances can still cause desync-esque animation effects.
Popo can constantly spotdodge while only Nana attacks, makes them almost invincible lol.
Running Blizzard is going to break some matchups, especially Falco's (more than it already is).
POPO getting grabbed causes Nana's whole "constantly Belay" thing. Not sure what else does, it's only happened to me when Popo's been grabbed so far (Edit: It's happened to me kinda randomly when not being grabbed, still not sure what else causes it). I believe we may be able to reset Nana after this if we can somehow make her come in contact with the ground, haven't pulled it off though. Edit: It's not possible to make her touch the ground, the game thinks she's offstage, slightly under one of the stage's tetherable ledges (chosen seemingly at random, possibly the most nearby one? it tends to change as you move around..), which is why she is constantly in her falling animation/trying to belay.
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Additionally, the opponent will TELEPORT (!) when trying to footstool Nana during this state. How this works depends on the stage, but generally either they will footstool real-Nana and warp offstage (happens on FD), or can footstool fake/offstage-Nana and warp onto real-Nana (Yoshi's Brawl). Occasionally I've also gotten the former to happen on stages where the latter happens by hitting them at the same time they footstool Nana, they warp offstage and then take knockback from the hit. Both teleports have advantages, former obviously puts them in a really bad position (sometimes insta-killing if Nana gets footstooled while she's jumping out offstage), latter can be taken advantage of by doing an attack with Popo and hitting them as they warp onto Nana.
Nana tripping is completely nullified, although she goes through a few animation shenanigans after a trip.
Dying doesn't reset the stickiness state.
Easy way to unstick: Belay or Squall.
Quick/Widely Practical way to unstick: ?
Other Ways to unstick: Nana footstooling opponent.
Easy way to stick: Get hit (lol).
Getting hit during this makes it look like only Nana is on the playfield haha (until you do something or their idle animations desync).
From what I can tell so far (not extensively tested it yet) the only thing the opponent can do to get you out of this is to force you into a situation where you need to recover. Edit: Grabbing Nana and throwing her away from Popo unsticks them.
There are more things I took note of that I can't think of atm, I'll post again later.
EDIT: Moar shiz.
For some reason Nana will randomly jump out of the stickiness, which will cause her to turn 90 degrees but otherwise be controllable normally (not sure how it affects hitboxes though), when Popo faces left Nana is facing the camera, when Popo is facing right Nana faces away from the camera. I've had it done sometimes when Popo's been grabbed, and also when engaging an opponent (taking hits/giving hits). She also jumps extremely high like a footstool. Theory: Nana is footstooling Popo.
Other fun stuff you can do: Midair Taunting, Popo is holding the ledge while Nana is taunting or doing other actions in midair. Moonwalking, Nana is dashing forward while Popo rolls back. Midair Walking, not sure what's going on but I'm guessing only Popo gets hit by a move so Nana stays in her standing animation while sticking to him, allowing her to "walk" in midair as he's launched.
Also grounded Blizzard doesn't face both directions anymore while stuck.
Oh boy, how I love new toys...