Many real-world systems are connected in a complex directed network, such as food webs, social, or neural networks. Spreading and synchronization processes often occur in such systems, and understanding the percolation transition (formation of a giant connected component) is key to controlling these dynamics. However, unlike in the undirected case, this had not been understood in directed networks with realistic nonrandom architectures. We provide a universal framework in which the percolation threshold for networks to be strongly connected (every node to be able to reach every other) can be analytically predicted on any real-world network and verify this on a diverse dataset. This explains why many real, dense networks are not strongly connected, in contrast to random-graph theory.