  We describe a particular class of pairs of quantum observables which are
extremal in the convex set of all pairs of compatible quantum observables. The
pairs in this class are constructed as uniformly noisy versions of two mutually
unbiased bases (MUB) with possibly different noise intensities affecting each
basis. We show that not all pairs of MUB can be used in this construction, and
we provide a criterion for determiniing those MUB that actually do yield
extremal compatible observables. We apply our criterion to all pairs of Fourier
conjugate MUB, and we prove that in this case extremality is achieved if and
only if the quantum system Hilbert space is odd-dimensional. Remarkably, this
fact is no longer true for general non-Fourier conjugate MUB, as we show in an
example. Therefore, the presence or the absence of extremality is a concrete
geometric manifestation of MUB inequivalence, that already materializes by
comparing sets of no more than two bases at a time.
