A subspecies of musical

Across The Universe and Mamma Mia! are unusual in that they are musicals (complete with production numbers) which were built around a bunch of pre-existing songs by a particular artist, and those songs were not written with this story -- or indeed, any story -- in mind.

It seems an interesting creative challenge to take a body of work written over the course of an entire career and find a story that links it all together, making the inclusion of each bit of music seem organic and natural. While I have no intention of seeing Mamma Mia!, ever, I found Across The Universe interesting and impressive; but mostly I like the idea of the challenge of writing something like that. I wish there were a body of well-known songs in Lusternia that my character could write a play around in a similar way; he's already written one musical romantic comedy with production numbers, and I'm very pleased with how it turned out, but I probably wouldn't do another unless I had a hook like this to make it new.

What I'm wondering, and if you know the answers post comments here, is this: is there a name for that particular subspecies of the musical genre? Can you list a few other examples of it?

To be clear, I know that even operas in Mozart's time and earlier were stitched together from librettos written by others, sometimes earlier and sometimes incorporating pieces that weren't originally written to be one thing. And I know there are many variety-show musicals which integrate pieces from different artists or different times in the careers of artists, as a sort of medley. And I'm familiar that in rock operas like Tommy, many of the songs were dug out of previous works and shoehorned into the work to pad things out. But what I'm looking for is specifically this challenge of taking a set of songs from a single artist but which were all written independently of one another, with no particular story in mind but their own, and then weaving them somehow into a single, coherent story.