Complete problems or alternative definitions of the complexity class NP^GI? Definition: Let $GraphIso$ be the decision problem whose input is a pair of undirected graphs $(G_1, G_2)$ and the output is true if and only if $G_1$ and $G_2$ are isomorphic.

Definition: Define $\textbf{GI}$ to be the class of all decision problems $L$ for which there is a polynomial-time Turing reduction from $L$ to $GraphIso$.

Definition: Let $\textbf{NP}^{\textbf{GI}}$ be the class all decision problems which can be decided by a polynomial-time non-deterministic Turing machine with access to an oracle for a problem which is in $\textbf{GI}$.

My question is then: is this class known? Is it equivalent to a more standard complexity class? Is there any know complete problem for this class?
[Summarizing the answers in the comments by Ricky Demer and Josh Grochow.]

$\mathsf{NP}^{\mathsf{GI}} = \mathsf{NP}^{GraphIso}$, since these are nondeterministic poly-time Turing reductions, which can be simulated by the oracle machine (as detailed e.g. here). 

The bounded-time halting problem for non-deterministic machines with a GraphIso oracle is complete for this class, by the usual proof.

Under standard derandomization assumptions (in particular, any that imply $\mathsf{AM} = \mathsf{NP}$), $\mathsf{NP}^{\mathsf{GI}} = \mathsf{NP}$, since under such assumption we have $\mathsf{GI} \subseteq \mathsf{NP} \cap \mathsf{coNP}$, and the latter is low for $\mathsf{NP}$.