TAT are pass-through shares of ASICMINER provided by "ThickAsThieves" hence the TAT abbreviation. Each TAT share is worth 1/100 of a normal Asicminer share or any other pass through share of Asicminer. Since you have to buy "whole" units of a share, this smaller unit allows people to buy portions of asicminer without having to pay 2+btc for a piece of the company.
The other versions of asicminer such as AsicMiner-PT or G.Asicminer are all the same shares of AsicMiner stock, just provided by a different person.
burnside is the provider for asicminer-pt and deadterra is the provider of g.asicminer.
Buying a direct share of asicminer establishes a contract with you and freidcat (owner of asicminer)
Buying Asicminer-PT share on establishes a contract with you and burnside, where burnside has a contract with freidcat.
Buying TAT-Asicminer share establishes a contract with you and thickasthieves, where thickasthieves has a contract with freidcat
and etc.
Thanks for helping!