Carbonate mud represents one of the most important geochemical archives for reconstructing ancient climatic, environmental, and evolutionary change from the rock record. Mud also represents a major sink in the global carbon cycle. Yet, there remains no consensus about how and where carbonate mud is formed. Here, we present stable isotope and trace-element data from carbonate constituents in the Bahamas, including ooids, corals, foraminifera, and algae. We use geochemical fingerprinting to demonstrate that carbonate mud cannot be sourced from the abrasion and mixture of any combination of these macroscopic grains. Instead, an inverse Bayesian mixing model requires the presence of an additional aragonite source. We posit that this source represents a direct seawater precipitate. We use geological and geochemical data to show that "whitings" are unlikely to be the dominant source of this precipitate and, instead, present a model for mud precipitation on the bank margins that can explain the geographical distribution, clumped-isotope thermometry, and stable isotope signature of carbonate mud. Next, we address the enigma of why mud and ooids are so abundant in the Bahamas, yet so rare in the rest of the world: Mediterranean outflow feeds the Bahamas with the most alkaline waters in the modern ocean (>99.7th-percentile). Such high alkalinity appears to be a prerequisite for the nonskeletal carbonate factory because, when Mediterranean outflow was reduced in the Miocene, Bahamian carbonate export ceased for 3-million-years. Finally, we show how shutting off and turning on the shallow carbonate factory can send ripples through the global climate system.