Failing the task but fulfilling the intent

In Burning Wheel we become well practiced at describing a successful task but failing the the intent. What does the opposite look like in play?

To search the room for Queen Anne’s secret communications with her brother, Philip IV.
Your task is that you start opening drawers. Intent, to find the letter without making a disturbance or being caught. To fail with a successful task is to find a hidden compartment in a drawer in Anne’s desk as Mme Combalet enters the room.

The example given in the book for failed task and fulfilled intent is:

*Due to a bad seal, you inhale some vapors of mercury during the operation. You have a crushing headache for 1d6 days and suffer a -1 to all rolls for the duration.” The player creates the recipe they were after, but at a cost to themself.

It looks like success with a condition without saying as much. Ruined gear, suffering relationships, loss of status. It’s that sort of thing, isn’t it?

As I understand it, you don’t necessarily have to add a condition. Any form of complication is possible here.
The philosopher of the book’s example could aswell succeed at creating the recipe but setting a curtain on fire while doing so or angering their neighbours with the foul vapors that arise from the kettle. Anything that indicates that the task failed is appropriate, I guess.

We could argue whether the second half of the sentence The player creates the recipe they were after, but at a cost to […]. is binding if you fail the task but fulfil the intent, but I would argue that any complication is possible, so that the rule would be something like You fulfil the intent, but […]. This would also reflect the design history of the game which at least names Blades in the Dark and Moldvay D&D as inspirations. Both games rely on such make-up complications.

You challenge her suitor to a duel to prove your love to her and win her heart.

She tragically falls in love with you as you lie bleeding.

You’ve achieved your intent while failing the task.