The field of computer science is in a bind: on the one hand, computer scientists are increasingly eager to address social challenges; on the other, the field faces a growing awareness that many well-intentioned applications of algorithms in social contexts have led to significant harm. We argue that productively moving through this bind requires developing new practical reasoning methods for those engaged in algorithmic work. To understand what such an intervention looks like and what it may achieve, we look to the twentieth century evolution in American legal thought from legal formalism to legal realism. Drawing on the lessons of legal realism, we propose a new mode of algorithmic thinking— "algorithmic realism”—that is attentive to the internal limits of algorithms as well as the social concerns that fall beyond the bounds of current algorithmic thinking. Algorithmic realism is a practical orientation to work, and thus will not on its own prevent every harmful impact of algorithms. Nevertheless, it will better equip engineers to reason about the sociality of their work, and provide a necessary first step toward reducing algorithmic harms.