When Billy Corgan appeared as a surprise guest on Howie Mandel’s podcast in January to talk to Bill Burr about the small, but not impossible, chance that they were half-brothers, fans were divided.

At this point, even Billy Corgan isn’t sure what is real life and what is fantasy. Since the Smashing Pumpkins singer appeared on Howie Mandel’s Does Stuff podcast in November and revealed that his ...