The goal of image reconstruction is obtaining an image of the object of interest from indirectly measured, and typically noisy, data. Mathematically this is formulated as an inverse problem, where the forward operator models data acquisition. In practice, not only the data are noisy, but also the forward operator is often not perfectly known as it may involve imperfect calibration measurements or simplified models. In this talk we present an approach to inverse problems with imperfect forward models that relies on partially ordered functional spaces - Banach lattices. A typical example of such an inverse problem is image deblurring, where the blurring kernel is not perfectly known. Failure to acknowledge errors in the blurring kernel may lead to reconstruction artefacts. These artefacts can be eliminated by modifying the reconstruction algorithms to account for errors in the operator. We discuss typical features of the partial-order-based approach in connection with total variation regularisation, such as loss of contrast, and show how the concept of debiasing on model manifolds can be used to overcome some of these issues.