After loosing the WW1, the Austro-Hungarian Empire was meant to be dismantled. Charles of Habsburg, opposed to this decision, was ablel to get a referendum. Everyone in the old Austro-Hungarian boundaries is allowed to participate. The question ? Do you want to remain under Hasburg monarchy.
What would happen ? Would Charles remain king/duke/whatever somewhere ?
Okay, looking for some limits.
I'll assume universal male suffrage? There is the precedent that woman had at some point been allowed to vote in the AH Empire, but at this point this privilege has been rescinded when the Kurien gave way to a more universal male voting right. Well universal for over 21 IIRC.
How will boarders be drawn? What administrative unit will be used here? Gemeinde, Kreis/Bezirk, Land? Even then how would one deal with the potential enclaves? Since post WWI the idea was clear boarders among ethical lines that didn't exist...
What about the territories given to the Italians, Romanians and Serbians? They do have the distinction of actually having fought as a nation in the Entente during WWI. Now I know the precedent of the referendum in Kärnten, but that was a relative small area...
Then there is the question of Poland, though they had less influence.
(German-)Austria, Czechoslovakia and Hungary of course were created only out of former AH territories, so they'd probably be created depending on the referendum.
Also you'd have to explain how they convinced the Entente to allow this referendum. Since Karl would have to convince them, nobody else would have the influence to make the referendum not just happen, but be accepted.
Then it comes down to other questions: What kind of Constitution would this new Habsburg Empire/Archduchy/Whatever follow? If it is one heavily weighted against the monarch I could see more people supporting it. Then there is the question of languages. And yes, that'd have to be in there. I'd assume the most likely way is ratifying the constitution with this referendum and the territorial question being second to this. That then most likely happen on the level of Länder, so pretty high level. And there I'd assume support would be pretty limited...
I'm just going to make some predictions without knowing clear limits and assuming everything is up for grabs. Also a relative weak monarch and still a constitution that promised multiculturalism, but can't shake the german overtones... Also kinda assuming that the OTL boarders are kinda known as a possible plan.
Starting in the West: South Tyrol proper would stay Austria, Trente probably won't, but I'm not that certain about that one.
Austria proper will stay "Habsburg" though Vienna might not vote for it, with their strong socialist base there.
There'd be a mess of conflicting votes in the mixed areas of Bohemia and Moravia. Mixed and German areas more likely to vote "Habsburg". From what I can remember a "Habsburg" Austrian Sliesa might also happen, annoying both Czechs and Poles.
I can see the parts of Galicia with strong Ruthenian and Jewish populations voting for "Habsburg" and against Poland.
Slovakia will probably be a mess as well. The only thing I'd think I'd win is a prediction that Bratislava might just go "Habsburg". The Hungarian influenced areas on the other hand probably won't follow.
Neither will Hungary proper. I know Horthy kept up that fiction OTL - but in this scenario I'd assume they'd drop Karl quickly.
There will be a mess in the Romanian area. I can see some pretty big "Habsburg" enclaves in there, mainly where large German minorities are present, but also against the trend Hungarians who do not want Romanian rule. "Better the devil you know" and all that.
I'm not certain about the SHS and rest-Italian areas. If any of them go "Habsburg" it's only because they are anti-Italian or anti-Serbian, not because of a large love to the Habsburg.
All in all you'd get a country that looks somewhat like a tattered carpet, with maybe worse enclaves and boarders than the Indian/Bangladeshi boarder was.
While OTL politicians shouted that Austria isn't viable because it's too small, I predict that this Austria, despite being bigger and more populous might actually be less viable than the OTL one...