Iterators are a common pattern in python. A trivial example is:

example = iter(("apple", "banana", "cherry"))
next(example)

(yes, of course, one does not need to treat this specific case as an iterator, it is just a minimal example)

How does one effectively work with iterators using the ExternalEvaluate / ExternalFunction in Mathematica?

Some strategies that do not work:

Strategy 1: ExternalObject

The tuple iterator gets returned as an ExternalObject, but then there is a type conversion error when sending it back to python to apply the next() function to it:

(* create a python session *)
session = StartExternalSession["Python"];

(* define the iterator, returning a ExternalObject *)
ex = ExternalEvaluate[session, "iter((\"apple\", \"banana\", \"cherry\"))"] 

(* define the next function *)
next = ExternalFunction[session, "next"];

(* but...applying the next function returns an error *)
next[ex] 
(* Error Message: WLFunction' object is not an iterator *) 

Strategy 2: PythonObject

The PythonObject resource function seems like it should help. Iterators define a __next()__ method that we should be able to call, but this does not seem to work:

(* define the PythonObject *)
ex2 = ResourceFunction["PythonObject"]["iter((\"apple\", \"banana\", \"cherry\"))"]

(* ex2["__next__"[]] *)
(* Failure: Message: name __next__ is not defined *)

(* ex2["next"[]] *)
(* Failure: Message: tuple_iterator has no attribute `next` *)