Tjyylli

In the story I created, the Kingdom of Oneirou has 3 Moons--Artemis, Rahu, and Zorya--all of which have their own independent lunar cycles.

However, it's officially stated as early as Chapter 2 or 3 that, once a year [each], all 3 Moons' phases are synchronized to Full Moon and New Moon. Let's say, for the sake of argument, that all 3 (while operating on their own separate lunar cycles) still follow the same 30~ days to complete one full cycle.

If we pretend that Oneirou still has the same 365 days = 1 Year like Earth, is this possible? If both events are not possible within a 365 day period, then how long would their year have to be extended to account for this?

You can get orbital resonances like those exhibited by Ganymede, Europa and Io around Jupiter.

an orbital resonance occurs when orbiting bodies exert a regular, periodic gravitational influence on each other, usually because their orbital periods are related by a ratio of small integers.

However, that doesn't match what you are asking.

all 3 Moons' phases are synchronized to Full Moon and New Moon

when a conjunction happens, it's either Full Moon or New Moon or whatever phase it is, but it cannot be more than one in the same cycle. Moreover, for the Ganymede/Europa/Io system there is never a triple conjunction.

all 3 (while operating on their own separate lunar cycles) still follow the same 30~ days to complete one full cycle

if they have the same orbital period, they orbit at the same distance. So, either they are the same body, or they are going to become one very soon. As you see from the resonance, the periods are different.

To have more bodies sharing the same orbit, there must be one significantly more massive than the others, and the smaller one have to be in the lagrangian points of the large one. But then they won't have a simultaneous phase.

This might fit your description: a 1:3:5 resonance.

Near moon: new on day  3 9 **15** 21 27, full on day 6 12 18 24 **30**.



Middle moon: new on day 5 **15** 25, full on day 10 20 **30**.



Far moon: new on day **15**, full on day **30**.

So: not all planets are having all their special phases on the same day, but there are days they're all having the same special phase.

I'm just a lowly software developer and don't know if a 1:3:5 resonance would be stable.

It's a question of interpretation whether a cycle need have one time around, or could be allowed more. For instance the full "Otto Cycle" in an internal combustion engine is actually two revolutions. Various climate cycles and orbital cycles, precessions of the earth's axis etc. are discussed in terms of many years.

Your mention of the 30 days period is under the rubric of "for the sake of argument" so I'm thinking it may not be central to your question.

You say they all have the same 30 day cycle, but separate cycles? They must then have separate phases, like a 3-phase electric generator.

https://en.wikipedia.org/wiki/Three-phase#/media/File:3_phase_AC_waveform.svg

Half the moon is lit at any given time.

In a new moon, the lit half is facing fully away from the planet (or close enough the sliver is pretty minimal). For this to happen monthly, this requires the moon's orbit be nearly coplanar with the planet-sun orbit.

If the moon is entirely coplanar, you'll get a solar eclipse every month. So the orbit may be slightly off coplanar like Earth's moon; that will make new moons common, but solar eclipses rare. *But they aren't really new moons then, since there is a thin wedge of lunar surface exposed to the sun and visible to the planet; they're more like 95% new moons. You'd need to declare a threshold of "good enough".

In a full moon, the lit half of the moon is facing "nearly fully" toward the planet, so the sun, planet and moon are in a row, in that sequence. But not quite that way; after all, that would be an lunar eclipse!

Again, the orbit may be slightly off coplanar, and it would need to be if you don't want a lunar eclipse every month. But again, *they aren't really full moons then; they're more like 95% full moons. Again a "good enough" is required.

"All full moons" mean all the moons are on the far side of the planet from the star, in the "good enough" zone.

"All new moons" would mean all the moons are between the planet and the star, again in the "good enough" zone.

We can't say definitively what's possible on your world because you have to define what you call "good enough". But possible? Sure. If you declare 10% of a moon's arc to be "new moon", then unless some other mechanic occurs controlling the moons' motion, you'll have triple new moons 0.1% of the time and triple full moons 0.1% of the time.