When multiple genes axe used in a phylogenetic study, the result is often a collection of incompatible trees. Phylogenetic networks and super-networks can be employed to analyze and visualize the incompatible signals in such a data set. In many situations, it is important to have control over the amount of imcompatibility that is represented in a phylogenetic network, for example reducing noise by removing splits that do not recur among the source trees. Current algorithms for computing hybridization networks from trees are based on a combinatorial analysis of the arising set of splits, and are thus sensitive to false positive splits. Here, a filter is desirable that can identify and remove splits that are not compatible with a hybridization scenario. To address these issues, the concept of the distortion of a tree relative to a split is defined as a measure of how much the tree needs to be modified in order to accommodate the split, and some of its properties are investigated. We demonstrate the usefulness of the approach by recovering a plausible hybridization scenario for buttercups from a pair of gene trees that cannot be obtained by existing methods. In a second example, a set of seven gene trees from microgastrine braconid wasps is investigated using filtered networks. A user-friendly implementation of the method is provided as a plug-in for the program SplitsTree4.