After completing the final Memory sequence, you cannot access the present. Be sure that you exit the Animus at least once to find an artifact.
There are a total of 5 artifacts to be found in present-day Monteriggioni, but you only have to gain one for the achievement. Collecting the rest will offer no extra benefits. Pick any of the 5, or just follow the one chosen for the video.
Claudia’s Record Book- After exiting the Sanctuary, turn directly around, and its in the room off to the right.
Maria’s Feather Chest- After exiting the Sanctuary, head up to the top of the villa, and its in a hole in the roof.
Mario’s sword- Head south of the fighting ring, and head right. The sword is beneath a tree in a rock.
Ezio’s Belt- Upon picking up the sword, head straight down the alley and turn left. Head up the wall at the end of the alley, and the belt is in the corner.
Medici Cape- Head East along the main wall until you see the Church. Head inside, and the cape is inside.
Check out this video for the location of Maria's Feather Chest:
I realized after completing the game that you needed to get this before then but I was planning to replay the game anyway so it's not a big deal. (I thought it was actually part of the storyline)
And DHc FiReHawK I haven't gotten the Desmond thing yet but I believe its only a skin like the outfits you can unlock through Uplay it doesn't allow you to play as Desmond outside of the animus unfortunately.
@9,23,&30, you cannot get this achievement the moment you first leave the animus, there is a certain sequence you must play to before u can get them i just dont know which one as i havent gotten there yet. I also checked all 5 locations after fixing the power lines like #9 said and they still werent there. I hate collectibles like this where you have to wait for the to magically appear like in ACII the feathers didnt appear until a couple of dna sequences. DUMB.
After sequence 2 i hit start and there was an exclamation mark next to the option "leave the animus" and like #5 said when i started to leave the sanctuary Lucy told me to put on an earpiece and watch. This is the absolute soonest i saw the exclamation mark, i checked every time i looked at my map, so i'm sure this is the soonest that you can get this achievement.
So I had to start a new game and play up until the crew is all set up inside the Villa in the present, and then immediately upon beginning sequence 2 you can finally press start during the first cinematic and choose "Leave Animus". As you try to leave, Lucy will tell you get back before dawn and to equip the nearby watch and earpiece which Desmond will do automatically. Climb to the top roof of the villa and as you approach the locker on the roof after using the lift, the achievement should pop.
Also, it took me about an hour to reach sequence 2, skipping all cinematics that allowed me to (press start and choose "Skip Cinematic"). Lastly, I can confirm that this achievement is indeed part of the base game, and does not require you own any of the DLC.