Financial markets are often fragmented, introducing the possibility that quotes in identical securities may become crossed or locked. There are a number of theoretical explanations for the existence of crossed and locked quotes, including competition, simultaneous actions, inattentiveness, fee structure and market access. In this paper, we perform a simulation experiment designed to examine the effect of simple order routing procedures on the properties of a fragmented market consisting of a single security trading in two independent limit order books. The quotes in the two markets are connected solely by the routing decision of the market participants. We report on the health of the consolidated market as measured by the duration of crossed and locked states, as well as the spread and the volatility of transaction prices in the consolidated market. We aim to quantify exactly how the prevalence of order routing among a population of market participants affects properties of the consolidated market. Our model contributes to the zero-intelligence literature by treating order routing as an experimental variable. Additionally, we introduce a parsimonious heuristic for limit order routing, allowing us to study the effects of both market order routing and limit order routing. Our model refines intuition for the sometimes subtle relationships between the prevalence of order routing and various market measures. Our model also provides a benchmark for more complex agent-based models.