I don’t think it’s possible in real life. What you shown is a drawing. Maybe people are confused and one country is looking at the southern cross and the other country is looking at the false cross. It’s kind of like why north hemisphere compasses won’t work correctly in the Southern Hemisphere. But the compass will always point to north correctly in both hemisphere. You can make a compass with a bottle top and a paper clip. That will point correctly to the North and it’s opposite to the south. But for some reason in the Southern Hemisphere you need an additional part that points to the South. This makes no sense. If your going opposite of north you will be going south and someone in Africa going south will not end up in the same place as someone in South America going south. Unless one uses a Southern Hemisphere compasses. Which sounds like it gives a false southern direction to make all points going south end up in the same place.
So a person can be in Africa and one in South America and both looking south and thinking there seeing the southern cross but actually be looking at 2 different sets of stars.
Two separate issues there - compasses and stars.
Let's deal with compasses first. You are right, some compasses (not all though - some are universal) are optimised to work in particular hemispheres, or even particular latitudes of each hemisphere. This has nothing to do with north or south though, and everything to do with dip - the downward angle of the earth's magnetic field, which varies with latitude. To compensate for the dip, northern hemisphere compasses are counterweighted on the southern end of their needle, and vice versa in the southern hemisphere. Use one in the wrong hemisphere and it will still point north/south, and it will do so correctly, but its performance will be poor as it will be trying to dip down at an extreme angle.
As for the stars, I'm afraid that just a ridiculous claim. Firstly, as I've said elsewhere today in another thread, the issue that we've illustrated with the extreme case of different continents is equally applicable, albeit not as obvious, for two observers just a few hundred miles apart on the same continent. If you and I met in Australia, we could look south, then look up at an angle equal to our latitude, say 30 degrees, and we would be able to see Sigma Octantis. Then we could get in our SUVs and drive East and West respectively, keeping that star on our left or right as we get further apart. Go for an hour or two in opposite directions and then look at the same star. It will still be orientated true south for both of us, and yet the according to the monopole FET map we are now each facing in two slightly different outward directions. How can we be looking in a slightly different outward, radial directions and yet looking at the same star? You've brought in this idea of different stars, but that makes no sense at all - where, when and how do these different stars appear? Where is the join? How do the various constellations retain their continuity and conformance to all of the different star charts? It can't be mistaken identity - that doesn't align with centuries of successful celestial navigation across the southern hemisphere.