The iOS application shows the current date using the Tranquility Calendar, a proposed solar calendar reforming the Gregorian calendar made by Jeff Siggins in the article "Lunar Timekeeper: A Special Calendar for the Space Age" in the July 1989 issue of Omni Magazine.
The iOS application provides some informational text, and the ability to convert dates from the current calendar to the Tranquility Calendar and from the Tranquility Calendar to the current calendar.
The watchOS application shows the current date using the Tranquility Calendar, and a variety of complications doing the same.