In your last criterion, you are essentially asking for a "natural" problem that is nonrecursive, recursively enumerable, and is not complete for the recursively enumerable sets. Post proved the existence of such problems in Recursively enumerable sets of positive integers and their decision problems, for many-one reductions. Friedberg and Muchnik proved this also holds for Turing reductions, in separate papers Two recursively enumerable sets of incomparable degrees of unsolvability (solution of Post's problem, 1944) and On the unsolvability of the problem of reducibility in the theory of algorithms. Whether these are "attractive" is probably determined by whether you like nonconstructive arguments. For a clear and self-contained exposition of these results, see Kozen's book Theory of Computation.
So this is only a partial answer, and it would still be nice to exhibit a real problem with intermediate degree.