Quantum systems achieve objectivity by redundantly encoding information about themselves into the surrounding environment, through a mechanism known as quantum Darwinism. When this happens, observers measure the environment and infer the system to be in one of its pointer states. We study the emergence of objectivity whenever the Hamiltonian of the system and the interaction Hamiltonian between system and environment do not commute, a condition where it becomes difficult to identify pointer states, and the use of methods such as the predictability sieve becomes necessary. We show in an explicit qubit model to what degree the noncommuting evolution allows for the emergence of objective states, and show that a definition of pointer states based on spectrum broadcast structure is more natural in this context, and coincides with the one based on the predictability sieve for commuting evolutions.