  Given integers $a_1,a_2,a_3$, there is a complex rank $3$ topological bundle
on $\mathbb CP^5$ with $i$-th Chern class equal to $a_i$ if and only if
$a_1,a_2,a_3$ satisfy the Schwarzenberger condition. Provided that the
Schwarzenberger condition is satisfied, we prove that the number of isomorphism
classes of rank $3$ bundles $V$ on $\mathbb C P^5$ with $c_i(V)=a_i$ is equal
to $3$ if $a_1$ and $a_2$ are both divisible by $3$ and equal to $1$ otherwise.
  This shows that Chern classes are incomplete invariants of topological rank
$3$ bundles on $\mathbb CP^5$. To address this problem, we produce a universal
class in the $tmf$-cohomology of a Thom spectrum related to $BU(3)$, where
$tmf$ denotes topological modular forms localized at $3$. From this class and
orientation data, we construct a $\mathbb Z/3$-valued invariant of the bundles
of interest and prove that our invariant separates distinct bundles with the
same Chern classes.
