REQUIREMENTS
This achievement requires you to destroy all 5 caves without losing your crown to the greed. You can have you crown fall on the floor as long as the greed doesn't pick it up. I haven't tested to lose one crown in coop and reforge it with the other character. However, you can't lose it during the explosion of the cave (more on that in the guide). Players have reported for other achievements that when done in coop, only the first player receives the achievement. I suppose it is the same for this one.
GENERAL STRATEGY
I recommend you to speedrun the game. It may seem strange, but the faster you complete this, the easiest it will be. In top of that, you will encounter less bugs and glitches, and lose less time if you somehow fail. Even if you don't have a friend playing this, play this in coop mode with 2 controllers. It is MUCH easier than solo because you have twice the coin capacity and the ability to give orders / watch the situation from 2 different places. There is no drawback for doing this in coop by the way. I chose the griffin as my main mount, as it can graze anywhere even in the caves. Therefore it is extremely easy to escape the bomb detonation. It can also repel greed with it's wings (double tap any trigger), to protect your knights and or builders when you expand far away. I stayed with the basic horse with the second character but you can use anything, such as a unicorn if you want (island 5). Just stay away from the bear and the drake. The first one is garbage, the latter has an interesting power which we won't need and is very slow.
Island order and pick ups:
Island 1 -> Island 2 (get the 8 gems, get the stone upgrade) -> Island 3 (get the builders statue for 3 gems, and if you want take the bakery hermit as well. Leave the island with 6 gems) -> return to Island 1 (get the archer statue and the griffin for 6 gems) -> Island 4 (get the iron upgrade, clear cave) -> Island 5 (clear cave, you may want the unicorn for the second character) -> return to Island 3 (clear cave) -> Island 2 (clear cave. In my successful run, winter came while I was here) -> Island 1 (clear cave and pop achievement)
WHAT CAN RUIN A RUN
Bugs. There are a lot of bugs in this game, and they will probably never be patched. Most common game breaking bugs are:
* the game will freeze upon loading a save and switching island. If you are online while it happens, you will be forced to delete BOTH local and cloud saves. It means to restart from scratch (all 3 saves will be erased)
* knights won't follow you in the cave. Needless to say that destroying all cores will be a chore even with archers and the griffyn.
* Buildings will despawn upon returning to a previous island, and none will be built no matter what you do (expand, build walls...)
* The forge won't spawn at all, despite having all requirements met (iron walls, iron castle, enough space). Please note that the forge may spawn very far from your base, or on the dock side, which are both annoying for this achievement.
*building a wall next or after your forge may despawn it forever. I had a wall ready to build just where you can buy swords. I accidentally built it and the forge disappeared.
By going offline before sailing to another island or entering the cave, you can delete the local save in case you have a bug. Then return online, resync the save, redo your gameplay up to the point where you went offline, and go offline again.
Poor RNG
* You may have a portal right next to your camp, and your archers will attack it during the day because they will wander too close to it. This will spawn greed during the day, so expect to lose a lot of archers every day
* Vagrant camps can spawn very far away, starting from island 3. I had 2 camps in island 5, near the dock and the cave. Even without a bakery, this is manageable in coop, but very annoying.
* As mentionned earlier, the forge can be tricky to spawn. Official requirements are: iron walls, iron castle and enough room for the building itself. However, sometimes it won't spawn on the cave side, or close enough to your base. This will make you lose valuable time, expanding iron walls here and there.
Mistakes
* Losing time will make the run much harder. If breeders or too many floaters spawn, they may break your defenses and steal your crown. My strategy is based on being fast enough to never see breeders, and very few floaters. You won't need to build a single archer tower, or a single wall except when you want to spawn the forge. Don't destroy any portal dock side either.
* Coming back to an island without money can allow the greed to overtake your poor defenses.
* Don't forget to graze with the griffin in the cave before detonating the bomb. Else, you might run short on stamina.
STRATEGY IN DETAIL
Island 1
Day 1: Skip the tutorial. Connect 2 controller and enable split creen.
Take the money with one character and start camp for 3 coins. The other character should run on one side of the island hoping to get coin chests. You will need 7-8 vagrants in total so hire them when you encounter a camp while exploring. The first character should spend his coins on: recruiting some vagrants, building lvl 1 walls, clearing a few trees on each side without despawning the merchant or the vagrant camp. Come back in the camp before the greed attacks. Hire another builder. One character should leave before dawn (right after the attack) and scout the other side of the island, again for coin chests.
Whenever you explore, take note which side has the dock and which has the cave on each island.
Day 2: while character 2 is exploring, use your first character to collect the merchant money and pay him 1 coin. Recruit a third builder and start building the boat. 3 builders is enough to build the boat and return to the camp in one day, but if you send them to late, you can always bribe the greed with a few coins to protect your builders.
Day 3: Collect more money from the archers and the merchant (pay him again), and ring the bell during the afternoon. Hop on the boat with many coins on each character and sail to island 2.
Island 2
Day 4: Get a character to start the camp, recruit both vagrants, buy 2 bows and 2 lvl 1 walls on each side. The other character should explore one side of the island. Save before crossing a portal in case it activates while you are on the other side, and collect the gems chests. Activate the stone upgrade if you see it. You won't need extra vagrants here, don't upgrade the camp or the walls. Just clear some trees to allow rabbits to spawn.
Day 5: Scout the other side of the island (get coin and gem chests, and activate the stone upgrade if it is on this side) and build the boat with the other character. You will need some coins to buy the other half of the boat pieces. I managed to leave the island during day 5, day 6 is ok. Remember that we don't build anything apart from the lvl 1 walls and we don't recruit anyone apart the 2 initial vagrants here.
Island 3
I will not list the actions day per day because you should know the basics: the less money you spend, the faster you can go and the easier the greed is. The idea is to prepare the camp as usual here, while sending the other character on each side of the island. The explorer should have 3 gems and some coins to activate the builder statue. Keep collecting gems until you have a total of 6 (we need 6 for the next island). Discard all the extra gems, we won't need them. Make sure you clear trees for the rabbits to spawn, and use your money on the boat. This will take more time, I left the island at day 13 but I lost time upgrading camp to stone. I recommend instead to leave as early as possible. Sail back to island 1.
Island 1
Here we only want to rebuild walls to lvl 1 if they are destroyed, activate the archer statue and grab the griffin with your main character. Rebuild the boat and sail to island 4. I left day 16.
Island 4
Now the real game begins. Start the camp for 3 coins, and upgrade it twice more for free walls the day you arrive. The other character should help clearing trees and exploring the island. The next day explore the other side, clear trees and keep upgrading your camp. When you have enough money, pay for the iron upgrade, and then upgrade your camp to iron. Now, start clearing more trees on the cave side, and expand with one iron wall, preferably around 2 screen away from the camp. Meanwhile, keep recruiting vagrants and hire archers and 2 squires on the cave side. Mak sure you have enough archers because when they tag along a squire, they stop hunting. If the forge spawned, upgrade both squires to knights and pay them coins. When you have some money on both characters, buy the bomb. Wait for a greed attack to end, and the move both characters towards the cave. DON'T pay the knights 4 coins through the wall, they will automatically escort you and the bomb for free. Move towards the first portal and wait for the archers to destroy it. With the archer statue, it is a breeze. Once destroyed, keep moving towards the cave and destroy both remaining portals. If a retaliation wave happens while you are on your way, simply stop and let your knights deal with it. They can handle anything apart from floaters but they won't appear that early in the game. Once you reach the cave, pay to enter. Make squre both characters are close enough, else you won't be able to pay (exactly when you want to sail in coop, both characters need to stand close to the boat). In the cave, take your time defeating each greed core. Leave character 2 near the entrance. Escort the bomb all the way to the detonating spot. Graze with your griffin, and only then you can detonate the bomb. Run to the entrance (where the other character is waiting) and leave with both characters at once. Congratulations, the greed will never spawn again on this island! Now you only need to get more money, rebuild the boat and sail to island 5. I sent the bomb day 26 and left day 31.
Island 5
We use the exact same strategy than for island 4. The only difference is now the island is a tad larger and vagrant camps can be very far away. You will need to leave the camp to recruit vagrants as soon as the greed wave ends, and sometimes you will need to repay them a second time because they wouldn't make it to the camp alive. Apart from that, there is no difference, apart from the cave having 5 cores instead of 4. I didn't lose more than a couple of coins in total in this cave, you have A LOT of room for error with knights. I sent the bomb day 39 and left day 41.
EDIT: on my second run, I was more efficient and I sent the bomb day 32. As you can see, sticking to the essential pays off. I also purchased the unicorn since there was more time before winter. You can get gems directly in island 5, you'll need 4 to mount this with your second character.
Island 3
Now it's only a matter of clearing the first 3 islands. Make sure you sail with enough money to build stone walls the day you arrive. I did not bother upgrading any archer tower, just take note that floaters may appear during retaliation waves. I sent the bomb day 47 and sailed day 48. Please note that winter comes day 49 in game, so make sure you anticipate to absence of economy unless you hire pikemen.
Island 2
Clear the cave and leave. I did that after winter, hence the time wasted: I sent the bomb day 70 and left day 71. I did not encounter a single floater or breeder while I waited for winter to pass.
Island 1
Even here during blood moons, I didn't encounter a single floater. So clear the cave and enjoy your achievement.
I am pretty sure that we can go faster, possibly clearing all 5 caves before winter (EDIT: my fastest run was 50 days, so barely into winter). I lost some time here and there, but the general strategy worked and this is what matters: don't waste time upgrading and building, just rush the caves :) Feel free to leave comments to improve this guide and share your own strategies!
BONUS
Each travel made in coop helps you towards the achievement for 50 coop travels. You can also continue on that save file for the 5 lighthouses achievement. Indeed, when a cave is destroyed, the greed will never spawn again on that island. Portals don't defend themselves, and there is no retaliation wave anymore, so you can easily send a couple of knights to clean the dock side.