Egg fragments are items that can be used at the Stables and pieced together in a 3x3 table in order to hatch a Monstie.
An egg fragment can have any combination of pattern, color, and position, and rarity. Also, you can only hold 100 egg fragments at a time, excluding any already placed in the egg tables. You also won't be able to place any egg fragments into the tables if your stables are full.

These can be ignored for the most part, since they aren't necessary at all for raising strong monsties. Creating an egg from fragments is required for a Rider Card Achievement, though, if you're aiming to 100% the game.

Egg fragments can be found by doing any of the following:

- Opening item chests in the Labyrinth of Ordeals
- Gathering at rare bone piles
- Participating in battles with other players

Rare bone gathering spots appear metallic bronze or gold in the original game, and dark brown in the remaster.

When winning against another player in battle, you'll receive 3 egg fragments, which will match the genus of their 3 Monsties.

If you have someone to play with, trading wins with each other is a much more efficient way of getting egg fragments compared to gathering them. You can speed up the process by fleeing at the start of each battle. And if you're looking for a specific egg fragment pattern, then have your opponent bring Monsties that match it. There isn't any way to influence the colors or position of the egg fragments, though.

All egg fragments have a pattern that matches a genus of Monstie. This determines the species of the hatched Monstie.
When you hatch an egg with multiple patterns in it, the game will pick one of the nine placed fragments, and then use this to decide the genus. After that, it picks a random species from said genus, and that's what hatches.
For instance, if an egg has 8 Amphib fragments and 1 Fly Wyv fragment, then it will have an 8/9 chance to hatch a Zamtrios (the only Amphibian monstie in the game).
It's also worth noting that your story progress affects which monsties can/can't be hatched this way.

Egg fragments can come in red, blue, and green, which correlate to ATK, DEF, and HP.
If you line up three egg fragments of the same color, you'll get a bingo. This will give you one hatch bonus in whichever stat that color bingo corresponds to.

Just like with hatching eggs from dens, a hatch bonus in ATK is 1 point, DEF is 2 points, and HP is 3 points. The maximum amount of hatch bonuses is 8, which also happens to be the largest amount of bingos possible in the egg fragment table.
In order to get a Monstie with all 8 hatch bonuses, you need to have all nine egg fragments be the exact same color as each other.

When finding egg fragments or receiving them after a battle, there's a chance you'll wind up with a rare one. These will hatch Monsties with extra open gene slots.

When you hatch an egg from only the usual egg fragments, the hatched Monstie will generally have the default/lowest amount of gene slots that species has. But if you use a rare egg fragment, then the hatched Monstie will have at least one extra gene slot comapred to what it normally would have.
Each rare egg fragment contributes one additional open gene slot.

This also means it's a waste to make an entire egg out of rare egg fragments, since you'll be using up several of them to no effect. The lowest amount of open slots something like Aptonoth can have is 4, which means using more than 5 gold fragments on any Monstie genus is also a waste.

There isn't any reason to go out of your way to find these, though. It's much easier to clear some floors in the Tower of Illusion to get a Stim item for the genus of Monstie you're trying to hatch.