  We show that completeness at higher levels of the theory of the reals is a
robust notion (under changing the signature and bounding the domain of the
quantifiers). This mends recognized gaps in the hierarchy, and leads to
stronger completeness results for various computational problems. We exhibit
several families of complete problems which can be used for future completeness
results in the real hierarchy. As an application we sharpen some results by
B\"{u}rgisser and Cucker on the complexity of properties of semialgebraic sets,
including the Hausdorff distance problem also studied by Jungeblut, Kleist, and
Miltzow.
