Henri III as King of both France and Poland?

I was reading through this thread, and it made me wonder why Henri III didn't just try to be King of both France and the Poland in personal union. He could have spent most of his time in France, where he would actually have power, whilst the Polish nobles might well have been happy to have an absentee monarch. It's not as if territorially non-contiguous personal unions were unknown at the time - look at the ex-Burgundian territories inherited by Philip II of Spain, or indeed the empire of his father Charles V, which was pretty much founded on personal unions.

Would this have been possible, and what effect would it have had on European politics, with Habsburg and Valois realms now encircling each other (Spain and the HRE encircling France, and France and Poland encircling the HRE)?
 
He did try. He maintained the claim for Polish-Lithuanian throne. Just the nobles of PLC were offended and they decided they need another monarch. The Commonwealth was in need of strong ruling individual at that time because it felt threatened by Muscovite expansion into Livonia and Courland. And at that time there were no magnates in PLC and the nobles were quite resonable citizens.
 
Still, Henry could arrange it better than escaping his kingdom under cover of the night. Couldn't he announce publicly in Poland that he is going to France for crowning, and nominate some steward to rule country in his place? Maybe try to make some arrangement with the Sejm how he's going to rule both countries? AFAIR after he escaped, he was sent letter by SEjm demanding him to show up in Poland again in certain period of time and he plainly ignored it.
 
The Poles aren't going to keep him as King if he leaves for France and Henri isn't going to stay in Poland following the death of his brother. The Poles didn't elect him to be an absentee landlord; they elected them to be their sovereign. Given the political situation in Poland at the time, the magnates don't want an absent king: this isn't a Poland that was failed state post-deluge of the late 17th century, or the Russian puppet of the 18th. You shouldn't project the views and desires of magnates from that century unto this century. Poland in this period was a fairly vibrant state. Yes, it was an elective monarchy: but they also wanted a strong king who would respect their institutions but also protect their borders. Henri was partially elected because of France's cordial relations with the Ottoman Empire. His election would secure breathing room on the southern border. He also promised to marry Anna Jagiellon (a promise he promptly disregarded), which was another reason his election was secured.

Henri wasn't that politically astute. We're talking about a man who had the Duke of Guise cut down on a whim in hopes it would weaken the league, only to toss France into another round of religious civil wars. The thought of being the King of France and Poland isn't even gonna click. Henri can stay King of Poland if Charles IX lives and begets heirs, or he can be the King of France when Charles IX dies. But he isn't going to maintain both, at least not for very long. The magnates practically regretted his election upon his arrival, and wished they had chosen someone with Jagiellon blood, as they later did when Stephen Bathory married the elderly Anna Jagiellon, and following his death Anna chose Sigismund Vasa (his mother had been Anna's sister, and thus he had Polish blood in his veins). Following the death of Sigismund II Augustus, the blood link of the Jagiellons was still important to the Polish magnates and hence I wouldn't say Poland was totally an elective monarchy. As I said before, part of Henri III's election was that he would marry Anna.
 
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