Regarding https://drafts.csswg.org/css-break/#break-decoration
Propose that a new value "bounds" for box-decoration-break.
"bounds" is a suggestion only, any string that conveys the intent.
The value intends to represent rendering the background-* properties such that they are applied to the bounding box of fragmentation boxes. The rendering would be limited to the fragmentation boxes.
E.g.

The border would be around the joined fragment boxes.
E.g.
| Scenario |
Example |
| Background |
 |
| Background clipped to text which has background fragmentation |
 |
Updated based on comment from @Loirooriol:
This suggestion applies mainly to the background portion of the box-decoration-break, so possibly this belongs as its own property/sub-property. E.g. box-decoration-break-background with values slice, clone, and bounds.