  We construct pairs of residually finite groups with isomorphic profinite
completions such that one has non-vanishing and the other has vanishing real
second bounded cohomology. The examples are lattices in different higher rank
simple Lie groups. Using Galois cohomology, we actually show that
$\operatorname{SO}^0(n,2)$ for $n \ge 6$ and the exceptional groups
$E_{6(-14)}$ and $E_{7(-25)}$ constitute the complete list of higher rank Lie
groups admitting such examples.
