Fracture network geometry is crucial for transport in hard rock aquifers, but it can only be approximated in models. While fracture orientation, spacing, and intensity can be obtained from borehole logs, core images, and outcrops, the characterization of in situ fracture network geometry requires the interpretation of spatially distributed hydraulic and transport experiments. In this study, we present a novel concept using a transdimensional inversion method (reversible jump Markov Chain Monte Carlo, rjMCMC) to invert a two-dimensional cross-well discrete fracture network (DFN) geometry from tracer tomography experiments. The conservative tracer transport is modeled via a fast finite difference model neglecting matrix diffusion. The proposed DFN inversion method iteratively evolves DFN variants by geometry updates to fit the observed tomographic data evaluated by the Metropolis-Hastings-Green acceptance criteria. A main feature is the varying dimensions of the inverse problem, which allows for the calibration of fracture geometries and numbers. This delivers an ensemble of thousands of DFN realizations that can be utilized for probabilistic identification of fractures in the aquifer. In the presented hypothetical and outcrop-based case studies, cross sections between boreholes are investigated. The procedure successfully identifies major transport pathways in the investigated domain and explores equally probable DFN realizations, which are analyzed in fracture probability maps and by multidimensional scaling.