Widespread iron oxide precipitation from groundwater in fine-grained red beds displays various patterns, including nodulation, banding and scallops and fingers. Hematite nodules have been reported also from the Meridiani Planum site on Mars and interpreted as evidence for the ancient presence of water on the red planet. Here we show that such patterns can autonomously emerge from a previously unrecognized Ostwald ripening mechanism and they capture rich information regarding ancient chemical and hydrologic environments. A linear instability analysis of the reaction-transport equations suggests that a pattern transition from nodules to bands may result from a symmetry breaking of mineral dissolution and precipitation triggered by groundwater advection. Round nodules tend to develop under nearly stagnant hydrologic conditions, while repetitive bands form in the presence of persistent water flows. Since water circulation is a prerequisite for a sustainable subsurface life, a Martian site with iron oxide precipitation bands, if one were found, may offer a better chance for detecting extraterrestrial biosignatures on Mars than would sites with nodules.