Hermitian–Walker 4-manifolds

A local description of proper Hermitian–Walker structures that are locally conformally Kähler, self-dual, ∗-Einstein or Einstein is given. This is used to supply examples of indefinite Einstein strictly almost Hermitian structures showing that an integrability result in [K.-D. Kirchberg, Integrability conditions for almost Hermitian and almost Kähler 4-manifolds, arXiv:math.DG/0605611] does not hold for metrics of signature (2,2)(2,2).