Recent research has shown the potential advantages of grid-forming (GFM) inverters regulating electric power systems, with many generation systems relying on power electronics. Addressing fault ride-through (FRT) conditions is an essential component of any control method of GFM inverters. This paper suggests an FRT scheme tailored for GFM inverters that takes into consideration current limits in distribution systems. The proposed adaptive control strategy addresses limitations in existing solutions, particularly during unbalanced conditions. It incorporates an adaptive sequence current allocation to enhance the GFM performance and is compliant with negative sequence current injection during unbalanced faults. Time-domain, as well as hardware-in-the-loop simulations, were used to assess the proposed scheme, showing its efficacy under all fault events.