Ionization isomerism in coordination compounds Ionization isomerism is defined as complexes which have same empirical formula but ionize to different ions in solution.

Consider the complex $\ce{[Pt(NH3)4Cl2]I2}$. The other possible ionization isomers are:
$\ce{[Pt(NH3)4I2]Cl2}$ and $\ce{[Pt(NH3)4ICl]ICl}$.

But I have not come across the complexes similar to the last one with two different anions outside the coordination sphere.
So, do these sort of complexes exist?

Moreover, what can be the IUPAC name for this compound?

tetraamminechloridoiodoplatinum(IV) chlorineiodine seems wrong.
I think it should be - 

tetraamminechloridoiodoplatinum(IV) chloride iodide.

The rules of the naming of counter ions are pretty much the same as in general Inorganic Chemistry. I do know of a compound AgClI - named as Silver chloride iodide. So, I think the counter ions here - chloride and iodide should be named as by the nomenclature rules of anions i.e with an -ide suffix and alphabetically if there is the presence of more than one anion

Proof of Reference - Silver chloride iodide (NCBI PubChem)

EDIT: My teacher did give an example of it being a valid complex. I think it is not generally stated in the preliminary books of this topic to avoid confusion among students.I imagine it would be present in some more in-depth books of this topic.