well... split shaft gas turbines and turbo compounding in one housing is like i said a different story from a thermodynamic view.
in a gas turbine you have only RE-action turbine stages where the gas from the combustion chamber ALWAYS blows the gas in it and you have a homogenous gas flow.
in your construction you have one RE-action turbine which is driven by the combustion gas ( the turbine which charges the engine via boost) an the second compound turbine in some circumstages gets an AC-tion turbine. this happens when you like i said close the throttle and shift down and the engine screems up - closing throttles = no exhaust gas flow on the engine to one stage and simultany the socond, engine rpm dependent compound turbine, revs up like the engine. in this moment the turbine is not driven by the gases but by the engine itself and "sucks" air in the housing instead being driven by air ( exhaust gases) . in this moment you will get an immediate stall at the turbine blades, because the blades, like an airplane wing, cannot work in both airstream directions.
thats why everybody, when if all, build multiple turbochargers or compound systems downstream or parallel, but never in one housing- or, like our wright TC3350, use one system by super and the other by turbocharging.
it maybe complicated to describe it via a forum, but really, it will not work so.
it is right that in some turboprops and all turboshafts ( helicopter engines) one stage is the gas generator and the other connected to the gearbox for the prop/rotor, but here you have a very slim rpm margin in which the prop or rotor changes speed ( constant speed prop) and excluding the situation of a windmillimg prop when the engine failed ( autorotaion in helicopters) you never have a situation when the prop/rotor feeds the turine instead of being fed by it.
axial, centrifugal compressor stages... well, a story for itself. both sytsmems have advantages. it would be a story for itself. quickly said- an axial system has a better efficiency, a radial gives you more pressure from one stage.
so we build a turbocharger- axial gas driven turbine for good efficiency, centrifugal compressor for high boost pressure in one rotor stage...