The good news is that these problems are going to diminish substantially pretty soon.
Recall the Orthodox calendar, which is 13 days off from the Western calendar. The Orthodox calendar is getting more and more off, by about 3 days every 400 years. But some of the Eastern Orthodox churches have adopted the Revised Julian calendar, also called the Milankovic calendar. The Revised Julian calendar omits 7 century leap years every 900 years, so there are 218 leap years instead of 225 every 900 years. (The two centuries that remain leap years are those that are equal to 200 or 600 mod 900, i.e., 2000, 2400, 2900, 3300, etc.) This makes an average year length of 365.24222 days. By contrast, the Gregorian calendar omits 3 century leap years every 400 years, so there are 97 leap years instead of 100 every 400 years. This makes an average year length of 365.2425; the two calendars will thus diverge by one day every 3600 years, on average. The true tropical year length is 365.24219, so the Revised Julian calendar is even a bit more accurate than our Gregorian one.
As I said earlier, it’s one thing to come up with a more accurate system; it’s another thing to get people to incur the transition costs. The Wikipedia article says that, so far, the Revised Julian calendar “has been adopted by the Orthodox churches of Constantinople, Alexandria, Antioch, Greece, Cyprus, Romania, Poland, and Bulgaria (the last in 1963), called the New calendarists. It has not been adopted by the Orthodox churches of Jerusalem, Russia, Serbia (including the uncanonical Macedonian Orthodox Church), Georgia, Mount Athos and [...]