That’s a complicated topic. Dancer Cat Glover tells a story of a meeting between the two icons:

On the plane, he brought up Graffiti Bridge as something he wanted to make into a musical. He said, “It’s gonna be you and Madonna.” She was actually supposed to be the lead of the movie. [Glover describes how Madonna later flew to Minneapolis to discuss the project.] He said he was having a hard time with Madonna and would I come down? He couldn’t deal with her by himself, is what he told me over the phone. So I came down and Madonna and Prince started arguing over the script. She was funny. I liked her. They both started ragging on each other. Madonna said she didn’t like the script, and Prince said to Madonna, “Well, I don’t like your shoes.” That’s how it all started. I was sitting there going, “Oh my God, here we go.” So Madonna told Prince, “I don’t like your shoes, either. Look at ’em with those peace signs and zippers and shit all over ’em.” And Prince was saying, “What are you wearing? Are those shoes or boots?” ’Cause Madonna had on these cowboy-boot shoes. This is what happens when you get two big egos in one room. It was jokey and serious. They’re the same way.

After their collaboration, Madonna suggested that she and Prince go on tour together, under the banner title of The Royal Tour … The Queen and The Prince. However, Prince reportedly turned it down as he said that “the world isn’t ready for this, it’s too big.”