An interesting example of a luni-solar calendar is the Chinese Agricultural Calendar, which keeps track of lunar months and the seasonal year. A new month and a new year always begins on the first day of a new Moon, a dark Moon. A common civil year has 12 lunar months and is between 353–355 days long, while a leap year has 13 lunar months and has 383–385 days.
The seasonal year is divided into 24 ‘solar segments’ that are precisely aligned with the 12 signs of the zodiac, each of which is split in two. These 24 solar segments have descriptive names indicating agricultural events that occur at that time of year, like “Day and night are equally long” at the Spring equinox, followed by “It is warm and bright,” about 2 weeks later, then “Rainfall is helpful to grain,” and on and on like that.
This Chinese Agricultural Calendar is what the Romans likely would of come up with if they were more interested in agriculture than taxes.