Although I didn't inquire on my last cruise in April they have always advertised that you could request a cruise card as a replacement. We've always enjoyed the Medallion functions and have no reason to be incognito.
Actually you're tracked less on a Princess cruise ship with a Medallion then you would be at home, even without a cell phone in your pocket. Your car license plate is being scanned no matter where you go as well as cameras in every store parking lot and as you drive down the street.
Can't live without one any more. I'm addicted.