What I understand...
A tomographically complete set of operators is a set of operators such that measuring the probability distributions of an unknown quantum state over the spectrum of each of these operators allows you to write down the pre-measurement quantum state (of course, one needs infinitely many copies of the identically prepared unknown quantum state to carry this out). For example, for a two-dimensional Hilbert space of spin half particles, the spin operators in $x$, $y$, and $z$ directions form such a set which I can verify by explicitly writing a unique state consistent with a given set of probability distributions over the spectrum of these three operators.
What I'm looking for...
I'm not sure if I understand the mathematical conditions that I can write down for a set of operators to tell me if it's a tomographically complete set of operators or not. Intuitively, I expect it to be something like "a largest set of non-commuting operators" because such a set would give me all the information about the phases which would be hidden if I perform measurements over a commuting set of operators. But what is the precise mathematical definition/criterion for such a largest set of non-commuting operators?
Wikipedia says that a tomographically complete set of operators forms an "operator basis on the Hilbert space". I don't think I understand this statement, for example, the three spin operators constitute a tomographically complete set of operators but I can't write down $S^2$ as a linear combination of $S_x, S_y, S_z$ which is something I should be able to do if $S_x, S_y, S_z$ formed a basis for all operators in the Hilbert space, or so do I think. Or, is this not what an operator basis on the Hilbert space supposed to mean?