It allows each section of the engine (fan, LP compressor, and HP compressor) to spin at the ideal rpm for any given flight condition.
Thank you for an informative answer.
There is a clear advantage of, at least RB.199 vs. the F-404 - the former is/was shorter by almost 70 cm than the later. On aircraft where the engines hang in the breeze that might not be important, but on the military aircraft where every liter of volume matters, this will, well, matter.
RB211 versus JT9D as installed on 747-100s meant about 7% less fuel burned on the same flights.
I've looked at RB.199 and F404. The early 199 was heavier than the early 404 by about 65 lbs (~6.5%), while making a much lower
dry thrust (40.7 kN vs. 47.15 kN), and also having a much better specific fuel consumption by a large margin, like 1/4 lower.
The A/B thrust was the same, however, and the
404 has better SFC by a large margin - 1/4 lower.
The latest comparable marks, like the 404-GE-402, and the Mk.104 and 105 on the last of the Tornadoes show the greater weight of the 404, and lower weight of the 199 (??). Thrust slightly improved on the .199 (by up to 2.2 kN dry, 3.2 A/B), while it improved for the -402 by 6kN dry and by 7.5 A/B. Seems like the RB.199 have had less of a stretch?
Europeans (basically RR) didn't went with the 3-shaft design for the EJ200.
IMO: a 3-shaft design scores points on it's physical properties, mostly the length. On other properties, there seems not to be an advantage, especially in a military application where a lot of thrust is required, thus negating the benefits of the lower SFC at low regimes. Downside also might be that the 3-shaft engine is more complicated and more expensive.