Lunar calendars (based on the 29.5 days it takes for the moon to go through a full cycle, from new moon to new moon) lose about 11 days per year compared to the sun. Agriculturalists who need to plant at approximately the same time per year keep track of the sun's progress. Many farmers believe that seeds grow best during a specific phase of the moon and thus keep track of the moon's progress as well.
How do they do that? A logical way is to have 30-day "moons" (months), with a feast-day, or festival, at each of the sun's solstices and equinoxes. Another way is to have thirteen months of 29 days each, as well as multiple days that would be celebrated at the solstices and equinoxes. Both approaches are complicated enough that they often require religious or secular specialists to keep track of them and schedule the necessary compensation days to reconcile the moon with the sun from year to year.
The Tohono O'odham calendar is pictured above (if anyone knows the original source of this illustration, please tell me). The Akimel O'odham have similar "moons," with the names varying somewhat.
I've incorporated the solar and lunar cycles into my "Tales of the Watermasters" series. In a full year, called a "Turning" by the Watermasters, the solstices and equinoxes are all observed as sun-festivals, with feasting and ceremonial events for each one. The lunar cycle in the Watermasters world consists of 29 days roughly constituting a "moon"; these are named after events that would have occurred seasonally, such as the Moon of Saguaro Fruit and the Moon of Mesquite Leaves. There are 4 additional days at each of the sun-festivals, with 1 - 2 days inserted into the calendar when necessary to adjust the timing of the new moon.
Tohono O'odham calendar mapped onto solar 12-month year