<p>Nonlinear feedback shift registers (NFSRs) are used in many stream ciphers as their main building blocks. An NFSR is said to be observable if any two distinct initial states are distinguishable from their resulting output sequences, and it is said to be nonsingular if its state diagram contains only cycles. To avoid weak key attacks, stream ciphers must use nonsingular and observable NFSRs. With condition that NFSRs only output their first bits, Fibonacci NFSRs are clearly observable, but Galois NFSRs’ observability is more complex and is little studied. This paper continues the research on the observability of Galois NFSRs with relaxation of their output restriction. Resorting to the semi-tensor product based Boolean network theory, the paper first reveals the observability index’s lower bound for general Galois NFSRs and its upper bound for singular and nonsingular Galois NFSRs and further for nonsingular (sub)maximum-cycle Galois NFSRs. It then gives some necessary and/or sufficient conditions for the observability of general Galois NFSRs. Based on the disclosure of the characterizations of distinguishable initial states on a cycle, the paper finally provides some necessary and/or sufficient conditions for the observability of nonsingular (sub)maximum-cycle Galois NFSRs, helpful to the design of stream ciphers.</p>