Ionization isomerism in coordination compounds
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.