Multi-agent neural implicit mapping allows robots to collaboratively capture and reconstruct complex environments with high fidelity. However, existing approaches often rely on synchronous communication, which is impractical in real-world scenarios with limited bandwidth and potential communication interruptions. This paper introduces RAMEN: Real-time Asynchronous Multi-agEnt Neural implicit mapping, a novel approach designed to address this challenge. RAMEN employs an uncertainty-weighted multi-agent consensus optimization algorithm that accounts for communication disruptions. When communication is lost between a pair of agents, each agent retains only an outdated copy of its neighbor's map, with the uncertainty of this copy increasing over time since the last communication. Using gradient update information, we quantify the uncertainty associated with each parameter of the neural network map. Neural network maps from different agents are brought to consensus on the basis of their levels of uncertainty, with consensus biased towards network parameters with lower uncertainty. To achieve this, we derive a weighted variant of the decentralized consensus alternating direction method of multipliers (C-ADMM) algorithm, facilitating robust collaboration among agents with varying communication and update frequencies. Through extensive evaluations on real-world datasets and robot hardware experiments, we demonstrate RAMEN's superior mapping performance under challenging communication conditions.