We propose a toy model for self-organized road traffic taking into account the state of orderliness in drivers' behavior. The model is reminiscent of the wide family of generalized second-order models (GSOM) of road traffic, it can also be seen as a phase-transition model. The orderliness marker is evolved along vehicles' trajectories and it influences the fundamental diagram of the traffic flow. The coupling we have in mind is non-local, leading to a kind of ``weak decoupling'' of the resulting $2\times 2$ system; this makes the mathematical analysis similar to the analysis of the classical Keyfitz-Kranzer system. Taking advantage of the theory of weak and renormalized solutions of one-dimensional transport equations [Panov, 2008] which we further develop on this occasion in the appendix of the paper, we prove the existence of admissible solutions defined via a mixture of the Kruzhkov and the Panov approaches; note that this approach to admissibility does not rely upon the classical hyperbolic structure for $2\times 2$ systems. First, approximate solutions are obtained via a splitting strategy; compactification effects proper to the notion of solution we rely upon are carefully exploited, under general assumptions on the data. Second, we also address fully discrete approximation of the system, constructing a $\BV$-stable Finite Volume numerical scheme and proving its convergence under the no-vacuum assumption and for data of bounded variation. As a byproduct of our approach, an original treatment of local GSOM-like models in the $\BV$ setting is briefly discussed, in relation to discontinuous-flux LWR models.