Inspired by an old idea of von Neumann, we seek a pair of commuting operators X,P which are, in a specific sense, 'close' to the canonical non-commuting position and momentum operators, x,p. Difficulties with von Neumann's original idea (involving an alleged orthogonalization of the coherent states) are discussed. Here these difficulties are avoided by restricting attention to operators acting on density matrices which are reasonably decohered (i.e., spread out in phase space). Such operators could be of use in discussions of emergent classicality from quantum mechanics. Moreover, they may be used to give a discussion of the relationship between exact and approximate decoherence in the decoherent histories approach to quantum theory.