Scientific visualization deals with increasingly complex data consisting of multiple fields. Typical disciplines generating multivariate data are fluid dynamics, structural mechanics, geology, bioengineering, and climate research. Quite often, scientists are interested in the relation between some of these variables. A popular visualization technique for a single scalar field is the extraction and rendering of isosurfaces. With this technique, the domain can be split into two parts, i.e. a volume with higher values and one with lower values than the selected isovalue. Fiber surfaces generalize this concept to two or three scalar variables up to now. This article extends the notion further to potentially any finite number of scalar fields. We generalize the fiber surface extraction algorithm of Raith et al. [RBN∗19] from 3 to d dimensions and demonstrate the technique using two examples from geology and climate research. The first application concerns a generic model of a nuclear waste repository and the second one an atmospheric simulation over central Europe. Both require complex simulations which involve multiple physical processes. In both cases, the new extended fiber surfaces helps us finding regions of interest like the nuclear waste repository or the power supply of a storm due to their characteristic properties.