If you want an Anjanath egg, you will need to wait until you get to the third area in Monster Hunter Stories 2, Lolsoka. In this area, you will be able to find Anjanath via Rare and normal monster dens.
You can also find Anjanath occasionally outside of these monster dens, but your best chance of finding Anjanath would be to search these monster dens often.
Once you do find an Anjanath, make sure to throw a paintball at it, as this will increase the chances of the monster retreats. If you don't have any paintballs, you can get them from paint berries which can be obtained from various berry nodes you can find in any area of the game's world. Just find a berry node and keep farming it until you get paintberrys.
That said, it's crucial you get paintballs, as making the monster retreat will also increase your chances of getting an Anjanath, so make sure to make the Anjanath retreat.
You can do this by breaking all of its parts, such as the nose, but to break the Anjanath's nose, you will need to have a strong water weapon - we recommend a hammer or a great sword to deal the most water damage.
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