My pet tanking mini-guide, current as of September 2017 (most recent change that affects pet tanking is the May 2017 addition of the Su3 Heyoka armor set)
I. Attachments:
Standard tanking attachments (1200+ JP) below, using Soulsoother head and Valoredge body (this allows for better capacity of commonly used tanking attachments). Bolded are musts, and you should use those regardless of JP. Shoot for at least grinding out 100JP for the extra attachment capacity, though even a 0 JP PUP with decent gear should be able to tank a lot of stuff.
Strobe I & II
Flashbulb
Auto-repair kit IV (or the highest you can fit if less than 100/1200 JP)
Optic Fiber I & II (can drop OF1 if needed due to lack of JP)
Armor Plate III & IV (or the highest you can fit if less than 100/1200 JP)
Mana Jammer III & IV (or the highest you can fit if less than 100/1200 JP)
Last two spaces have some room to play with and aren't as important. Options include:
* Galvanizer - counter (I usually use this as a standard choice)
* Speedloader II - help prevent the puppet from breaking SCH SCs by waiting a little longer to try to WS when it has TP (usually causes the puppet to wait long enough to allow a SCH immanence self-SC to go off)
* One of the random 1-earth attachments: Barrier Module, Schurzen, Hammermill, Analyzer
* One of the water attachments: Heatsink, Steam Jacket, Resister
* Regulator - Absorb-Attri, need to use a dark maneuver to trigger it
* Resister I/II can be used (and could drop a Mana Jammer to make room if enfeebles are the concern and you're not as worried about magic damage), though unless you have a very strong Meva set otherwise (like from alluvion weapon/armor Pet:Meva+ augments), and perhaps a GEO using Vex on the mob, you might still be working with a very low resist rate (particularly with needing to mix in fire/light maneuvers), making it hard to build around this as a strategy.
II. Master Weapon:
You have a few options. Condemners (with pet DT augments), Midnights (path D), Ohrmazd, and even the lv75 Oberon's Sainti (with pet: DT-). Could even just leave on your DD weapon if you plan to punch anything, sometimes I just tank in my Kenkonken for that reason (and the overload suppression is something, I guess).
Personally, I use an Ohrmazd with pet DT-4%, Meva+20, and str/dex/vit+15, but it can be a costly hassle to augment alluvion skirmish weapons.
Path D Midnights are probably the most painless option to acquire, and have a nice mix of Overload reduction (not totally irrelevant when you have to spam fire on a newly activated puppet) and pet DT-3%, plus the maneuver effect+2 as a little bonus.
III. Master Armor:
For body/hands/legs/feet slots, three relevant sets:
1) Taeon with Pet:DT+4% is great, I pair mine with regen+3 and Meva+22~24 on all four non-head pieces and use this as my default tanking set.
2) Rao/+1 (path C: pet DT-/HP/acc). NQ has less DT than max Taeon, but the added HP can be nice. Honestly not super important for survival (but hey, more HP is more HP), but relevant for the higher HP pool for enmity purposes.
3) Heyoka (Su3 set, requires 500 or more JP): for the Enmity+ to establish initial hate, or to recover hate for any reason (including hate reset mechanics). Personally, I have a macro with 5/5 Heyoka and both enmity+ earrings, and I manually swap into that for initial engage and provoke/flash, and when needed (particularly as strobe/flashbulb timers are almost up in anticipation of the voke/flash, or any time you have to get a mob back after a hate reset).
Other slots:
Animator: Animator P (or +1) is a must. In addition to the shown attribute+ stats (which are in addition to the base stat+89 from ilevel119 Divinator), it gives a significant amount of puppet defense, evasion, meva, etc. which isn't reflected in the item description.
Head: Anwig Salade, DT-10%
Neck: Shepherd's Chain, though if you lack that you might try Empath Necklace, or just idle in master DT- neck
Ears: Enmerkar/Rimeice is ideal, and if you don't have those can mix in Domesticator's (mainly for the Enmity+5), Handler's+1 (but this can be useless if you're already capping PDT, Burana (regen).
Rings: Thurandaut/+1 if you have it, but it's not that big of a deal if you don't and you can cap PDT/MDT anyway without the slot. I actually traded mine back in for a ring for another job, and I don't miss it one bit. Aside from that, only Overbearing Ring provides any help for tanking (Pet:HP, both for survival and higher HP for enmity purposes). I'd generally avoid Varar rings to reduce pet TP as much as possible and avoid it screwing up SCs.
Back: Visucius's Mantle with Pet: Regen+10 (most important augment). Can do whatever you like with the other two slots, or nothing at all and it's still best for tanking. I tossed Master Eva/Meva on mine, as well as Pet:MAcc/MDMG (nothing relevant to tanking for that slot, and can double as best pet nuking/Macc cape).
Waist: Isa Belt
Legs: Foire +2/+3 are best pet:DT- in the slot, and are certainly a legitimate option, but I find that simply gearing for max DT isn't necessarily best, nor is it necessary. You might be capped anyway, especially for PDT, and even if you aren't capped a puppet tends to be so sturdy that you won't be in danger due to sheer HP anyway - the threat is almost always from nasty status effects or weird mechanics. Therefore, I prefer the slightly lower DT and additional perks of Taeon (Meva, Regen) or Rao (HP). Tali'ah Seraweels are another option (Pet:DT-4%), and while I'd go with any of the other Taeon/Rao/Foire first, Tali'ah will perform perfectly fine.
IV. Misc. Strategy Notes:
Puppets are at their best on a single target mob without major hate resets (since you're relying on Strobe/Flashbulb timers and the puppet is just going to use those when the timer is up). Holding multiple mobs is a real hassle, and is where something like a PLD will really have an advantage over you. Also, you'll want to try to avoid popping stuff where adds go after the popper (e.g., most Escha-Ruaun T2/T3 mobs). But if it's just a single mob without doom or other insta-kill mechanic shenanigans? Your little automaton buddy is nearly indestructable with practically no support required (except infrequent cures on the master, which you may be able to handle with /WHM).
You will find that with proper attachments and a standard Fire/Light maneuver setup (1-2 of each, mix in more light when you want more regen, fire when you want more enmity), your puppet will be in very little danger from direct damage for the vast majority of fights. Can occasionally have use for something like a Water maneuver for MDT or Resister too, or when Overdrive is up and you only need 1 maneuver for full effect. The real threat is nasty AoE status effects, which you'll want to handle with Maintenance and Repair (use AF shoes to remove more effects!). Don't forget that Overdrive makes puppets immune to the vast majority of status effects, which can be handy for some mobs that use obnoxious enfeebles. Doom is a straight up pet killer, can't remove it so your puppet will just die - making PUP a less than ideal tank for mobs that use frequent doom.
One of your biggest tanking challenges will be the puppet WSing when you don't want it to. Especially if you're relying on SCs from the backline, either SCHs or ranged DDs. Some ways to mitigate this:
(1) Use Tactical Switch when the puppet approaches 1000tp (have 0tp on the master) to swipe the puppet's TP. Take your weapon off quickly to zero out your own TP to prepare for next time. This is far from foolproof though, since the 3min timer often isn't enough to be ready to remove your pet's unwanted TP.
(2) Use Speedloaders to delay pet WSing and allow others to get their WS in first. Speedloader tends to make the puppet wait until the last moment of the SC window to follow someone else's SC, making this help when you can't get rid of the pet TP
(3) Avoid using excessive Pet:Acc+ gear (or pet acc food) when you can help it. Heyoka, Tali'ah, and Rao gear give the pet some extra acc that you might not want, so prepare for that (one more reason I like Taeon as a default). Same idea for other pet TP generating stats like Store TP or Double Attack.
(4) If all else fails and you know your puppet is going to WS, just warn your SC team. It sucks to have to waste a SCH's immanence charges, so if SCs are vital I'll let them know the puppet is about to WS and to wait.
A significant threat is the much squishier master dying to something like a huge range AoE or adds. Make sure you are either out of range of nasty stuff or someone is prepared to heal the master, and don't be afraid to swap into DT gear if you know you're gonna eat something like a big Meteor.
Subjobs: When I'm there solely to tank I'm usually /WHM for Cure IV (with capped potency set), and to toss out -na/erase or hastes on others as needed.
It's nice to be able to replace a dead puppet instantly with Activate or DEA without any weakness timer. Can even be an intentional strategy. For example, in our LS we like to do Kyou with two tanks - we let the puppet be main tank and just eat Unfaltering Bravado and die, PLD holds while the puppet is re-activated and re-establishes hate (heavy use of Fire maneuvers and my pet enmity+ set). No need for others to have to get into conal range that way.