I was thinking about this at lunch, and I think it doesn't matter how the timing works out, just as long as you're not trying to do anything 'simultaneously'.  As long as you agree that each tiny step of the resolution has to happen in *some* order, you're good.  You run into one of the following situations:

1) IO is taken, Control of Character switches as IO is no longer on it, IO is discarded.  Attempt to put IO on a character, but it's gone, so no resolution necessary.

2) IO is taken, put IO on your own char, Control of original char switches, IO is discarded.

I think that both of these interpretations are invalid since you have to back-track the resolution.

3) IO is taken, control of character switches, IO is discarded, attempt to put IO on a character, you can't because it's not in play, backtrack and invalidate the original IO being taken.

4) IO is taken, put IO on your own char.  Control of original char switches, IO is discarded - but wait, that means it can't have been applied, back track and say it can't be taken.

In order to come up with the situation where you can't take control of it, you have to sometimes completely resolve a card effect and other times not.  If you consider it in the terms of "having to completely resolve a card effect", then you have either of these two situations, that apply to the #1 and #2 above. 

1) Repo effect starts,  IO is taken, IO effect starts, control of char switches, IO is discarded, IO effect finishes, Repo effect finishes.

2) Repo effect starts, IO is taken, IO is put on own char, Repo Effect finishes.  IO effect starts, control of original character switches, IO is discarded. IO effect finishes.

In fact, reading that, I think #2 is probably right since #1 involves nesting. 

 Yeah, but don't forget that you can't take control of the attachment if you can't play it.  I think that because IO gets discarded when it is removed it can't be taken in the first place.

This is a good one that was partially clarified recently:

The Well - Story Card

Each player chooses one story card he has won (except this one) and shuffles it into the story deck. Then destroy all Day and Night cards in play.

Sample Scenario:  Player A has 1 story, Player B has 0.  Player B wins the Well story.  It has been clarified that the second part cannot happen because the first effect cannot completely resolve.  Can player B still force Player A to shuffle their story card into the deck even though he cannot complete the requirement?

As long as you have a character on your side of the board you always have a legal target for the stealing of Infernal Obsession. I take control of Infernal Obsession per Repo Man's ability. I detach IO to move it to a legal target on my side of the board. IO having been moved that first character immediately switches back to my control. Now I have at least two legal targets for IO, Repo Man and Bob. I don't think there is any discard.

Penfold said:

I'm on the same page here. IO is just discarded if you take control of the character IO is currently controlling but since it's detached before and the card requirements are reset, there is no reason for discarding it, if you can attach it to a valid target.

 Well you will always have one valid target when you start to resolve Repo Man, and then two when it comes time to reattach IO.

 My argument is not about a legal target, but rather that the IO card must be discarded, so cannot be attached.

How do you figure? It is only discarded if the character it is currently attached to changes controllers. Taking control of the attachment and detaching it would be functionally the same thing per the rules. That means IO is detached when the character comes back to you so its discard effect cannot resolve.

If this were AGoT I would guess it would be a different issue. Their moribund would make the whole process a bit easier to understand (though the concept of moribund itself is a little hard to wrap your mind around).

Well, think of it this way.  When does the "take control and attach to legal target" part take place?  If it all happens at once, and then you resolve the effect of IO, no problem.  You take control, attach it to your own.  Done.  Then you resolve IO's text, it get discarded.  No problem with the "cannot take it because we can't attach it" issue as we've already taken it.

The only issues comes up if you resolve some of the Repo ability, then resolve the IO ability, then return to resolving repo.  This is not allowed actually.

(v1.0) Simultaneous Effects
When card effects, passive abilities, or forced responses simultaneously affect multiple cards controlled by a player, all cards that are affected resolve in the order determined by the card’s controller, one at a time. The player must fully resolve each effect before the next effect takes place.

Whenever a card effect affects both players simultaneously, the active player resolves his effects first.

Whenever character or support cards enter or leave play at the same time, the controlling player chooses the order in which they enter or leave play. They are not required to follow the order in which they originally entered or left play.

I copied the whole rule here, but I don't think it really matters.  I've bolded what has to be the most relevant section.  So this is the very definition of multiple cards and passive effects triggering.  He picks Repo first.   Begins resolving it, and must fully resolve it.  Attachment leaves character one, and attaches to character two (Note:  Character one is not a valid target for the re-attachment since it is still controlled by the opponent.  Repo, however, would be a valid attachment target).  Repo is done resolving.  Now, IO must resolve, and must resolve fully.   The originally controlled character switches side, then IO gets discarded.

KallistiBRC said:

But that's just the case if you assume, that IO has some kind of memory, if I understand it correctly.

As I said before, if IO is re-attached to a new character, in my opinion it practically resets all past events and is treated like a new card in play.

Ahzrab said:

Nothing about the process would cause the IO to need a card memory (which the rules state do not exist) since the card has never left play.  When it switches to a new character, it still has to resolve it's passive effect. 

Consider cards like Chess Prodigy who even if they DO leave play, their effect remains until the end of the phase.  IO would have to be considered the same. 

 Nah. The card text on IO refers specifically to the card it is attached to. The second it is unattached it ceases to be a concern. There is no way for the rules to allow it to be discarded because of something happening to a character it had been attached to when the card specifically states is attached to.

The character immediately reverting control once IO is unattached or remaining under your opponents control until after Repo Man's effect fully resolves is another matter though. A case could be argued both ways. I need to check my copy of the FAQ about gaining control, but CoC does not have a moribund rule like AGoT does, so I'm not sure how IO being removed is solved in a rules way to allow it to remain under your opponents control with no game mechanic forcing it. An argument could be made that reverting control is part of resolving Repo Man's effect.

I do know that IO has no current rulebook or FAQ entry that would make it be discarded though.