Basically, you can run each window in a separate process (use the answer here to see how to implement JavaProcess which is used below):

public class Tiles {
   public static void main(String[] args) {
      LwjglApplicationConfiguration configForTiles = new LwjglApplicationConfiguration();
      TilePresets tilesWindow = new TilePresets();
      LwjglApplication tiles = new LwjglApplication(tilesWindow, configForTiles);
   }
}

Wrapper.java is the main entry point. It's where launching both windows occurs:

public class Wrapper {
   public static void main(String[] args) {
      // Launch mapWindow regularly 
      LwjglApplicationConfiguration configForMap = new LwjglApplicationConfiguration();
      MapMaker mapWindow = new MapMaker();
      LwjglApplication map = new LwjglApplication(mapWindow, configForMap);

      try {
         int res = JavaProcess.exec(Tiles.class); // Where the second window is shown
      } catch (IOException e) {
         e.printStackTrace();
      } catch (InterruptedException e) {
         e.printStackTrace();
      }
   }
}

Credit alert! I had a similar question and I found this solution somewhere, but I can't remember where. I'll post the source here when I find it.

Edit: Credit goes to the person from which I got the idea for this solution.