This page explains how to start and complete the Stranded on Eventide shrine quest.
How to Start the Quest
If you go to Eventide Island, which is in the southeasternmost part of Hyrule, in the Necluda Sea, this quest automatically begins. Be warned: all your armor, equipment, and items will be unavailable as soon as you set foot on the island. You can teleport away if you want your belongings restored. You also can't save while you're on the island if you haven't completed the shrine quest.
To get to Eventide Island, go to Chaas Qeta Shrine, pick up a Korok Leaf next to the palm tree if you don't have one, then stand on the raft, and swing the Korok Leaf in the direction you want sail. Go southeast to reach the large island.
How to Complete the Quest
Since you have no items now, be sure to pick up everything you find. There are crabs and lizards on the beach, mushrooms next to some of the palm trees on the beach, and you can pick up rocks to find crabs, lizards, and other things. There is a boat oar on the beach that you can use as a weapon, and there are also tree branches in the grassy areas.
If you upgraded the Sheikah Slate to add the Sheikah Sensor (see Locked Mementos for more information) you can use the Sheikah Slate to help you find weapons and food.
You can still use Remote Bombs, so do that if you are out of weapons.
There are some Bokoblins at a campfire on the beach. You can defeat them and take the fish cooking at the campfire, as well as the Boko Spears that the fish are on. Then use Magnesis to lift and drop the metal boxes, which contain fruit. Then open the treasure chest where the Bokoblins were. It contains a Traveler's Sword.
There is a bog in the northern part of the island with a tree stump in the middle that has a Woodcutter's Axe on it. You can use Cryonis to make platforms over the bog and reach the axe.
Find an Orb
Southeast of the northern bog on the island, there is a Bokoblin camp, and they have one of the three orbs that you need to find. If you don't have a bow, there is a cliff where you can glide onto the watchtower with the Bokoblin that has a bow. You can try to land behind this Bokoblin when it's looking away, and kill it to get its bow and arrows. Before you kill the remaining Bokoblins, you might want to to dodge their arrows and then pick up the arrows where they fall (you can take a picture of an arrow for the Hyrule Compendium and then search for arrows with the Sheikah Sensor to help you find them).
After you defeat the Bokoblins, pick up the orb and take it to the north end of the island, where there is a glowing platform. The water is too deep to walk to the platform, but the orb floats, so throw the orb into the water between you and the platform. Then you can use Cryonis to make a platform beneath the orb, or try to chop down the palm trees to make a log you can walk on near the orb to pick it up. Drop the orb into hole in the platform.
Find a Second Orb
South of the Bokoblin camp where you got the first orb, there is a Blue Hinox. It has an orb on its necklace. The easiest way to get this orb is to climb up the slope to the east of the Hinox (but not all the way, because there is an enemy camp there), then float on top of the Hinox and pick up the orb while the Hinox is still asleep, then run away. When you run far enough away, the Hinox will give up and stop chasing you.
The correct destination for this orb is on the glowing platform in the southwest hill of the island, but you will have to defeat the Bokoblins at the Bokoblin camp there first.
Find a Third Orb
The third orb is guarded in a large enemy camp on Koholit Rock, on the southeast part of the island. There are two Bokoblins on watch towers that you can try to shoot before they see you, if you have a bow and arrows. There are three Bokoblins around the fire, and a Blue Bokoblin near them. There is also a Blue Moblin.
After you defeat the monsters, you can take the orb that is on one of the watchtowers. The glowing platform has a big flat rock on it, but if you have any Octo Balloons from defeating Octoroks on this island, you can drop one on the rock to make it float away. Otherwise, you can use Stasis on the flat rock, and hit it with a weapon until the arrow turns red. When Stasis wears off, the rock should move out of the way and you can put the orb into the platform.
Visit the Shrine
After you put the third orb into the third platform, the Korgu Chideh Shrine will appear on top of the hill on Koholit Rock, on the southeasternmost part of the island. Your belongings will be returned, and you can save the game again.