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I'd like to add to the above solution that, despite the description for the achievement, you don't need to get both personal unions at the same time, you just need to have two personal unions at the same time. While Castille certainly works, I personally got my achievement in an Austria playthrough where I forced a personal union with Bohemia just as the game started by claiming the throne and declaring a succession war, and accidentally got one with Salzburg over a century later when their ruler died heirless.

At the start of the game, you can get a Hasburg member of ruling age on the Bohemian throne through a royal marriage immediately, and ensure a Hasburg prince (not of age) ends up on the Hungarian throne. Hungary will usually kill Hasburg prince when he comes of age (15), which will give you a cassus belli to enforce a personal union without the negatives for claiming the throne.

If you have any ideas on how I can improve this solution, I'd love to hear them! There may be something that isn't here that should be, or an easier way of doing this. I'm always open to suggestions. If you notice any typos or poorly phrased sentences, please let me know.