Hi there,
it is Zoldy-Z right here!
I am veteran League of Legends player and this my second guide. My first guide was 6 Bloodthirsters Troll Fiora... so I could not count that one.
Lets get to the guide itself!

Full Tank Anivia Support

This champion build is not so good in early game, when Anivia has -Magic Resist and -Armor from her passive, but in mid or late it can be very hard to kill even when is in Egg form. You can ask why to play Anivia as tank role. And answer is pretty simple: Anivia has very unique skills and spells and she is very squishy, so she can be killed very easily. BUT if you make her unkillable and she will be still able to use her super powerful skills, which is purpose of this build --> that makes her super strong support, even if she had some really big weaknesses.

Itemization is pretty easy, but there are still some very situational items, so lets get to it!

We can start with explaining the Final Build 1:

eye of the equinox

So, for this item explanation is very easy. We just need wards, that is super core for every support and then we need HP when you want to be a tank and building Ruby Sightstone and face of the mountain is just too much for your inventory space, so we can just combine it to Eye of the equinox

Ninja Tabi

This is just for that to have balanced Armor and Magic Resist in the build, but the passive of these shoes is very helpful in mid game skirmishes. It is good (mainly when playing with Vayne as our ADC) to build Boots of Swiftness.

Athene's Unholy Grail

The purpose of this item is very obvious. Anivia is f*cking mana hungry bird, so she need some mana regeneration and it also gives some Magic resist and Ability Power, which is kinda useful.

Sunfire Aegis

This item is questionable, but I think it is better option than Dead Man's Plate or Randuin's Omen, because it is very good for doing some more magic damage, because Anivia with this build does not give much damage.

Banshee's Veil

This item is just filling the tanky potential and giving the big resistance to magic damage.

Sterak's Gage

The use of this item is very cool. It can save proc of Rebirth by its passive. Or it can save Anivia from dying when in Rebirth. And it gives Attack damage, which is worth for autoattacking.

Now we should continue by items that are different in the Final Build 2:

Dead Man's Plate

I think, that this item is not as good as Sunfire Aegis, but it is very strong too. Mainly in combination with Iceborn Gauntlet it give Anivia pretty big slow on basic attack and that means that she is able to hit her Flash Frost much easier. And it also gives a nice movement speed to chase down enemy carries.

Mercury's Treads

These boots are quite good when is Anivia getting the Iceborn Gauntlet first, because it balances the Armor and Magic Resist and gives Tenacity to chase down enemy carries easier.

Iceborn Gauntlet

It is almost the same as the Dead Man's Plate (Autoattack slow), but it is even the replacement for Athene's Unholy Grail, because it gives Anivia a big mana sustain andd the same cooldown reduction. And when Anivia has the Sterak's Gage, so the Autoattack after spell gives even more physical damage.

Other situational items:

Thornmail In fact, this item is useful only against full Attack damage or almost full Attack damage. But very cool thing is that its passive work for Eggnivia (in Rebirth), so she can return the damage from Autoattacks too.

With Example full build 1 (at level 18 and without Elixir of Iron):

Health: 4072.6
Mana: 1296.04
Attack Damage: 136.22
Ability Power: 60
Movement Speed: 370 (units per second) (without Wanderer )
Health Regen: 46.41 per 5 seconds / 9.282 per second
Mana Regen: 40.8 per 5 seconds / 8.1 per second (all without Athene's Unholy Grail and Meditation )
Armor: 177.22 (without Legendary Guardian )
Magic Resist: 135.25 (without Legendary Guardian )
Cooldown Reduction: 20%

With Example full build 2 (at level 18 and without Elixir of Iron):

Health: 4027.6
Mana: 1796.04
Attack Damage: 136.22
Ability Power: 0 (Sad truth)
Movement Speed: 370 (units per second) (without Wanderer )
Health Regen: 46.41 per 5 seconds / 9.282 per second
Mana Regen: 20.4 per 5 seconds / 4.08 per second (all without Meditation )
Armor: 213.97
Magic Resist: 129.75

If you can use W (wall) well, so you can max it like this:
If the using or landing wall isharder for you, so you should do it like this:
Now, we will explain usage of her spells one by one:

Pros:

+ Very Hard CC
+ Quite unkillable
+ Fun to play
+ Can counter low mobility Champs very easily
+ Giving her ADC a lot of space for farming or doing plays

Cons:

- Very mana hungry
- Low mobility
- Hard to master
- Hard to last hit to share gold (can be mastered)

YES:
If you have any of these as your AD carry:

Vayne
- She is actually the best to Anivia support, because the combo:
Crystallize + Condemn --> Flash Frost --> Vayne's followup --> Brutal damage and kill in most of times.
But: It is not so easy to land the 2E combo against good players. So you have to be well synergized with your AD carry.

Kalista
- Her mobility allows her to jump through Anivia's wall and there is super combo:
When Anivia dies in skirmish, and she proc her Passive and Kalista uses her ultimate for the egg it drags the Eggnivia and she uses Kalista's ultimate then she (when used well) jump alive as Anivia (bird form).

Graves
- Graves is cool ADC for Anivia too, because his Q ( End of the Line ) is returning instantly (almost..=0.25 sec) after collision with terrain. And what terrain you can ask! And I would answer: Crystallize. This Combo gives super big burst of damage when combined with other Anivia's and Graves' spells. And Graves' Quickdraw can easily get him through Anivia's wall.

Lucian
- Lucian's mobility is insane (So Crystallize is not big problem for him when is badly used) and his The Culling is perfect with combination of Anivia's Flash Frost and Glacial Storm can make ultra burst of damage and in case of doing it to low mobility ADCs can mean kill.

Miss Fortune
- It is not so good combo for botlane as other ADCs above, but together, they can still do some really nice amount of damage. Their combo should be:
Make it Rain + Flash Frost + Bullet Time + Glacial Storm + other skills.

Corki
- Corki is very good to Anivia, because even if is wall passed for example by Flash he can still use Valkyrie. Anivia's CC power is very good in cmbination with Corki's big burst damage. The clear combo is just super obvious: Flash Frost + Phosphorus Bomb + Gatling Gun + other Anivia's and Corki's abilities.