Inconsistency in formal ontologies is usually addressed by computing repairs for the dataset. There are several strategies for selecting the repairs used to evaluate queries, with various levels of cautiousness and classes of computational complexity. This paper deals with inconsistent partially ordered lightweight ontologies. It introduces a new method that goes beyond the cautious strategies and that is tractable in the possibilistic setting, where uncertainty concerns only the data pieces. The proposed method, called C{\$}{\$}{\backslash}pi {\$}{\$}$\pi$-repair, proceeds as follows. It first interprets the partially ordered dataset as a family of totally ordered datasets. Then, it computes a single data repair for every totally ordered possibilistic ontology induced from the partially ordered possibilistic ontology. Next, it deductively closes each of these repairs in order to increase their productivity, without introducing conflicts or arbitrary data pieces. Finally, it intersects the closed repairs to obtain a single data repair for the initial ontology. The main contribution of this paper is an equivalent characterization that does not enumerate all the total orders, but also does not suffer from the additional computational cost naturally incurred by the deductive closure. We establish the tractability of our method by reformulating the problem using the notions of dominance and support. Intuitively, the valid conclusions are supported against conflicts by consistent inclusion-minimal subsets of the dataset that dominate all the conflicts. We also study the rationality properties of our method in terms of unconditional and conditional query-answering.