By interconnecting smaller chiplets through an interposer, 2.5D integration offers a cost-effective and high-yield solution to implement large-scale modular systems. Nevertheless, the underlying network is prone to deadlock, despite deadlock-free chiplets, and to different faults on the vertical links used for connecting the chiplets to the interposer. Unfortunately, existing fault-tolerant routing techniques proposed for 2D and 3D on-chip networks cannot be applied to chiplet networks. To address these problems, this paper presents the first deadlock-free and fault-tolerant routing algorithm, called DeFT, for 2.5D integrated chiplet systems. DeFT improves the redundancy in vertical-link selection to tolerate faults in vertical links while considering network congestion. Moreover, DeFT can tolerate different vertical-link-fault scenarios while accounting for vertical-link utilization. Compared to the state-of-the-art routing algorithms in 2.5D chiplet systems, our simulation results show that DeFT improves network reachability by up to 75% with a fault rate of up to 25% and reduces the network latency by up to 40% for multi-application execution scenarios with less than 2% area overhead.