Could there be any situation where two noncommuting observables are simultaneously considered?

Question

As physicist Robert Griffiths (one of the founders of the 'Consistent Histories' formalism) says:

"Two physical variables whose operators do not commute correspond to incompatible sample spaces, and in general it is not possible to simultaneously assign values of both variables to a single quantum system"

Note that he says in general, so there are special cases where this can be done. These special cases, as he says, is when although AB does not commute with BA, they may still have in the spectral representations projectors in common (or that commute).

But can you think of any extremely special or exceptional case (for example in the 'Consistent Histories' formalism) where two noncommuting elements with literally no projectors that commute could be considered simultaneously?

Answer 1

Two non-commuting operators can have a common eigenvector if this eigenvector is an eigenvector of their commutator with a zero eigenvalue. This can happen, for example, for two components of angular momentum.