Self-organization occurs in plasmas when energy progressively transfers from smaller to larger scales in an inverse cascade. Global structures that emerge from turbulent plasmas can be found in the laboratory and in astrophysical settings; for example, the cosmic magnetic field, collisionless shocks in supernova remnants and the internal structures of newly formed stars known as Herbig-Haro objects. Here we show that large, stable electromagnetic field structures can also arise within counter-streaming supersonic plasmas in the laboratory. These surprising structures, formed by a yet unexplained mechanism, are predominantly oriented transverse to the primary flow direction, extend for much larger distances than the intrinsic plasma spatial scales and persist for much longer than the plasma kinetic timescales. Our results challenge existing models of counter-streaming plasmas and can be used to better understand large-scale and long-time plasma self-organization. PUBLICATION ABSTRACT