Characters are defined using details to represent who they are, what they are like and who they are linked to. By default, details don't have any mechanical uses in the game apart from providing narrative liberties.
An attachment is a character detail that you can choose to temporarily lose to resist an incoming consequence. If you want to resist a consequence, but don’t want to make a determination roll, you can decide to break one of your attachments. Doing so cancels the consequence outright.
However, this can only be done when it makes sense narratively, and that particular attachment cannot be used until time is taken to repair or reacquire it.
To reacquire or repair an attachment, the PC needs to start a project progress clock and track it on their character sheet. The clock starts at 4 segments, but may have more segments if the attachment is complicated to repair, if it was heavily damaged, or if it is difficult to reacquire.
To make progress on the project, a character may only roll to repair a "broken" attachment when they have the appropriate amount of downtime.
Once the project is completed, the character is permitted to use the attachment in the fiction, and is allowed to block a new consequence by breaking it once again.