  The problem of finding provably maximal sets of mutually unbiased bases in
$\mathbb{C}^d$, for composite dimensions $d$ which are not prime powers,
remains completely open. In the first interesting case, $d=6$, Zauner predicted
that there can exist no more than three MUBs. We explore possible algebraic
solutions in $d=6$ by looking at their `shadows' in vector spaces over finite
fields. The main result is that if a counter-example to Zauner's conjecture
were to exist, then it would leave no such shadow upon reduction modulo several
different primes, forcing its algebraic complexity level to be much higher than
that of current well-known examples. In the case of prime powers $q \equiv 5
\bmod 12$, however, we are able to show some curious evidence which -- at least
formally -- points in the opposite direction. In $\mathbb{C}^6$, not even a
single vector has ever been found which is mutually unbiased to a set of three
MUBs. Yet in these finite fields we find sets of three `generalised MUBs'
together with an orthonormal set of four vectors of a putative fourth MUB, all
of which lifts naturally to a number field.
