  It is a crucial feature of quantum mechanics that not all measurements are
compatible with each other. However, if measurements suffer from noise they may
lose their incompatibility. Here we determine the critical visibility such that
all qubit observables, i.e. all positive operator-valued measures (POVMs),
become compatible. In addition, we apply our methods to quantum steering and
Bell nonlocality. We obtain a tight local hidden state model for two-qubit
Werner states of visibility 1/2. Interestingly, this proves that POVMs do not
help to demonstrate quantum steering for this family of states. As an
implication, this also provides a new bound on how much white noise the
two-qubit singlet can tolerate before it does not violate any Bell inequality.
