The paper you linked (<PERSON> et al.) is an experimental realisation of a game proposed in another paper (<PERSON> et al.), and this game is only able to be accurately explained with a complex quantum theory (or something more complicated, see below).
You're right that are complex-quantum-theory and real-quantum-theory mathematically different? is an entirely theoretical problem, and that problem is answered in the positive by <PERSON> et al. (As you can see from the previous discussions of you linked, this answer was suprising!) They proposed a spesific <PERSON>-like gate that will have one behaviour in complex-quantum, and a different predicted behaviour in real-quantum. They also include much greater theoretical discussion of their definitions, assumptions and implications, so I'd recomend this as the paper to read if you're interested in that.
The experimental qustion then answered by <PERSON> et al is is the real world then complex-quantum or real-quantum?. They run such a game and find that complex-quantum-theory is necessary to explain the experimental results.
What is meant by complex vs real quantum theory
<PERSON> et al begins with a clear definition of '(complex) quantum theory', containing the usual elements: 1. A physical system corresponds to a complex valued <PERSON> space $H$ 2. Measurements correspond to projection operators on $H$ 3. The liklihood of a given measurement outcome is given by the inner product of the eigenstate representing that outcome with the current state of the system; the <PERSON> rule. 4. Combining two systems is just a tensor product: $H_a \otimes H_b$
'Real quantum theory' is defined analagously, but with a real-valued Hilbert space in (1).
<PERSON> et al's game
They give a spesific game involding 'entanglement teleportaiton' which complex quantum theory allows but real quantum theory does not. Basically, it works by having A and B share a <PERSON> pair, B and C share a different <PERSON> pair, and B performing a Bell measurement on the two halves of the pairs they have access to, resulting (in complex quantum theory) in A and C now sharing a <PERSON> pair. 682
The entanglement between AB and BC has been 'teleported' to be between AC, despite A and C never interacting. Reproducing this in a real values quantum theory is not posible. Interestingly, they need a 'multi-party' <PERSON>-like game to separate real and complex theories; a regular 'single-party' Bell game keeps the same predictions in both. Still, multiparty games of this type (if not exactly this game) have been performed before.
<PERSON> et al impliments this game precisely on superconducting quanutm hardware. You can see in Fig2 they make 2 Bell pairs (labeled EPR), and then perform a bell measurements (labeled BSM). They show what we should expect is true for complex quantum theory. (If you're especially interested, I'll note that this game is quite easy to impliment on current superconducting hardware, and it would not be a stretch to reproduce it on IBM's quantum experience.)
Locality
Assumption (4) above is perhaps non-descript but important; it corresponds to locality of information in the theory, sometimes called 'Local Tomography'. <PERSON> et al notes that while their game distinguishes real and complex quantum theory with assumption (4), it is easy to construct a real valued theory that prefectly matches quantum theory by also violating (4) and introducing some non-locality. Examples they give include the path-integral formalism and Bohmian mechanics. To return to the game above, reproducing the result in a real quantum theory is only possible if you add some non-locality that lets you reproduce the necessary correlations between A and C.
We should then perhaps rephrase the overall claim that we can distinguish that quanutm mechanics in our world is either complex-valued or inherently non-local (in which case the using complex or real returns to being only a methamatical convenience). The second case is 'considered nasty by physicists', which explains the choice of definitions they use: 'real-valued quantum theory' is taken to mean 'real and local quantum theory', which has now been effectively falsified. 682