Complete problems or alternative definitions of the complexity class NP^GI?
[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 $\mathsf{NP}$.