When you first meet Yda outside of your apartment in Ul'dah, she has a full-sleeved tattoo that goes all the way up her arms. However, after she asks you to be her bodyguard at the Aguellala Shrine , she tells you that she's going to get rid of the tattoos. Then, when you meet her again in the Northern Thanalan, she has no tattoos.

Why did yda's tattoo disappear?

Firstly, the tattoos only disappear in quests where she tells you that she wants to get rid of them. The quest is called "Tattoos for Yda." There are two versions of this quest, one in level 30 and one in level 50. When I did this quest for the first time at level 50 , it was with Lyse and Y'shtola.

The quest was called "Tattoos for Yda" at level 50 .

Yda will say something like: "It's time to get rid of my tattoos..." or "I hope you don't mind but I'm going to remove my tattoos as we move forward." Once this happens, her tattoo will disappear.

After completing the quest and having Yda not display any tattoos , you receive a weapon called "Broken Tattoo" from her . This weapon has an item level of 60 and provides 20 attack , 10 accuracy, and 24 crit as main stats.

According to the Japanese FAQ page for FFXIV, there is no actual explanation for the disappearance of Yda's tattoos .

The second version of this quest is at level 30 . When I did this quest , Yda gave me some kind of "black water" to pour on her tattoos. It was called "Tattoos for an Artificer!"

After completing this version, there was no change in her appearance.

The only thing I can think of as an explanation for this inconsistency is...

Yda went to a different town between levels 30 and 50 . However, she is from Ul'dah so it's odd that she would go to Limsa Lominsa or Gridania. Perhaps she went to Ishgard? Maybe her parents sent her there to be a priestess?

At level 30 , Yda said she was going to get rid of her tattoos by submerging them in black water. Perhaps this is because Ishgardians do not like tattoos... but Ishgard didn't even exist when these quests were made so it's hard to say anything for certain. :)

What gives?

Well, it turns out that Yda's disappearance of the tattoos was just a continuity error by the game designers. And don't worry if you wanted to get one of your own - they have not disappeared forever  - they are just invisible when she's wearing her leggings.

That's right, she is actually still wearing the tattoos!  When you do finally strip her down to nothing but her tatted-up pasties at Godbert Manderville's behest (or, if you watched "that scene" with poor F'lhaminn), they are shown clearly on her bare skin.

So, while some have figured it out by noticing her tattoos while she's wearing the leggings, others still don't believe it.

Conclusion

There are several possibilities as to why Yda's tattoos appear and disappear. The most likely explanation is that she went to a different town between levels 30 and 50.

In the level 30 quest , there is no change in her appearance after completing the quest. This means it could have been a continuity error, something that wasn't noticed until FFXIV 4.0 when the quest was reworked.

At level 50 , when you receive the Broken Tattoo after completing the quest, Yda can't wear her leggings with them on. This is why they disappeared in patch 4.0 . However, this mistake/error has already been fixed and she can now display her tattoos properly when you do the quests in FFXIV 4.0.