Globally localizing a mobile robot in a known map is often a foundation for enabling robots to navigate and operate autonomously. In indoor environments, traditional Monte Carlo localization based on occupancy grid maps is considered the gold standard, but its accuracy is limited by the representation capabilities of the occupancy grid map. In this paper, we address the problem of building an effective map representation that allows to accurately perform probabilistic global localization. To this end, we propose an implicit neural map representation that is able to capture positional and directional geometric features from 2D LiDAR scans to efficiently represent the environment and learn a neural network that is able to predict both, the non-projective signed distance and a direction-aware projective distance for an arbitrary point in the mapped environment. This combination of neural map representation with a light-weight neural network allows us to design an efficient observation model within a conventional Monte Carlo localization framework for pose estimation of a robot in real time. We evaluated our approach to indoor localization on a publicly available dataset for global localization and the experimental results indicate that our approach is able to more accurately localize a mobile robot than other localization approaches employing occupancy or existing neural map representations. In contrast to other approaches employing an implicit neural map representation for 2D LiDAR localization, our approach allows to perform real-time pose tracking after convergence and near real-time global localization. The code of our approach is available at: https://github.com/PRBonn/enm-mcl.