Simulation frameworks for autonomous mobility-on-demand (AMoD) systems are currently implemented as round-based games, where the real-world simulation time effectively stops until the fleet operator and users finish their decision-making processes. As such, these decisions, which take time to complete in the real-world, are unrealistically treated as instantaneous in the simulation.This work presents a real-Time gaming framework for AMoD simulations that creates asynchronous processes to compute fleet operator decisions and delayed mode choice decisions of users (after they receive real-Time information), while continuously updating the system state, i.e. moving vehicles and performing boarding processes. The framework enables a multiplayer game with multiple AMoD operators using different policies and pricing strategies to compete against each other and public transportation for on-demand travelers.Two operator strategies only available in a framework with non-instantaneous decision-making processes are introduced. The first strategy involves forecasting the future system state when the operator decision will probably be applied. The second policy sends a vehicle to the assigned request even if the corresponding user has not confirmed service with this operator yet.The framework is applied to numerical experiments in order to simulate effects of non-instantaneous operator and user decision-making. As expected, results show that operator computation time becomes relevant for the number of served requests in case of impatient users who want to make their mode choice decision shortly after their request. The new operator strategies also show beneficial behavior in the tested scenarios.