In the present climate, tropical rain bands exhibit a bifurcated pattern, continuously forming along single intertropical convergence zones (ITCZs) in some regions, and along double ITCZs that straddle the equator in other regions. This bifurcated ITCZ pattern is studied in a Matsuno-Gill-type model forced by relaxation to zonally asymmetric surface pressure. The model includes an idealized Bjerknes feedback which couples surface winds and sea surface temperatures (SSTs) via oceanic Ekman balance. Consistent with observations, solutions in the limit of strong damping are explored. Two ITCZ bifurcation mechanisms are identified. First, in the viscous limit, ITCZs form along negative anomalies of the local Rossby number, which tend to occur near the equator for equatorial low pressure and off the equator for equatorial high pressure; this leads to a single ITCZ in the rising branch of zonal-overturning circulations and a double ITCZ that straddles the equator in the descending branch. Second, ocean upwelling produces an equatorial cold tongue with increased surface pressure, which reduces vertical winds and can lead to precipitation peaks that straddle the equator in regions of near-equator ascent. Consistent with observations, the cold tongue intensifies with increasing zonal SST gradients, and its base widens with weakened poleward SST gradients, modulating the zonal orientation of the ITCZs on either side of the cold tongue. Analytic approximate solutions in the viscous limit capture the emergence of the bifurcated ITCZ pattern, as well as the dependence of the bifurcated ITCZ pattern on zonal and poleward SST gradients Plain Language Summary Tropical rain bands lie along near-surface convergence zones where the lower branches of the meridional overturning circulation intersect. In the present climate, tropical rain bands exhibit a bifurcated pattern, continuously forming along single intertropical convergence zones (ITCZs) in some regions, and along double ITCZs that straddle the equator in other regions. Since the tropical overturning circulations in regions dominated by single and double ITCZs are inextricably linked, a theory of the zonally varying tropical rain belt requires consideration of the bifurcated ITCZ pattern (i.e., concurrent single and double ITCZs) as a whole. In this paper, a classical simple model of the tropical circulation is used to gain a conceptual understanding of the bifurcated ITCZ pattern. It is found that the bifurcated ITCZ pattern emerges in the limit of strong mechanical damping. In addition, when surface winds are coupled to ocean heat transport, an equatorial cold tongue emerges which modulates the intensity and regional characteristics of the bifurcated ITCZ pattern in a manner consistent with observations.