This achievement is unlocked by completing the Easter Islanders scenario. Upon completion of each statue you will be given the option to choose an outcome/effect of erecting the statue. To achieve perfect status, you must always choose the option on the left.
For those having trouble with this scenario, here tips based on what worked for me:
1. Pause the construction of the first statue. As they are all in order, this means they will not progress until you unpause this one.
1. Build a small marketplace underneath the distillery and use that as the basis for where you will refocus your town. Do not extend your population outside this area - this stops the wild growth and decline of population, and keeps food consumption steady. Build 2 pastures to make use of the animals found from the hunting lodge (also using dung as fuel saves you on timber)
2. Research in the following order: Education (the first line to get the 3 points), water reservoir, boiler room, houses, and then just start on the tech trees to get to the deep mine and deep quarry (or at least mine and quarry), and reinforcement for buildings.
3. Build a nursery home as soon as possible to shift older people out of those limited homes and keep the population moving
4. Do not waste stone and iron on unnecessary items until you have at least one quarry and mine setup (I put these north of town near the statues).
5. The disasters don't start getting aggressive/cumbersome until you erect the statues, so if you have done above you should be able to have a town full of approximately 200/100/50 in reinforced buildings, stable population and enough food.
Once you have the above, the rest is just a walk; resume construction on the statues, and keep improving the town as you like. (or just go do something else and periodically check back to turn the speed back up after a disaster strikes)
Notes:
If you experience a population decline, shut off the forester lodges first. you can occasionally just cull all trees from the map normally and they will regrow by the time you need more so you don't really need these anyway.
If you are also looking to do both this and the imperfect achievement (close) together, then I recommend saving the game when the workers are at 9000/9999 of erecting the sixth (second last) statue. (More on that in the related achievement).
Lastly, DO NOT under any circumstances remove either any of the statue resources (the stone, the ropes/logs and the display slab) or the distilleries - you can't get them back and without the distillery the people get miserable pretty quickly.