Summary: We consider open exponential networks with routing matrices that depend on a network state. A customer entering a node is either independently of other customers queued with probability that depends on the network state or instantly bypasses the node with complementary probability. After bypassing a node, customers are routed according to a stochastic matrix that depends on the network state and may differ from the routing matrix. Under certain restrictions on parameters of the model, we establish a sufficient ergodicity condition and find the final stationary distribution.