This paper studies the Rayleigh-Plateau instability for co-flowing immiscible aqueous polymer solutions in a microfluidic channel. Careful vibration-free experiments with controlled actuation of the flow allowed direct measurement of the growth rate of this instability. Experiments for the well-known aqueous two phase system (ATPS, or aqueous biphasic systems) of dextran and polyethylene glycol solutions exhibited a growth rate of 1 s-1, which was more than an order of magnitude slower than an analogous experiment with two immiscible Newtonian fluids with viscosities and interfacial tension that closely matched the ATPS experiment. Viscoelastic effects and adhesion to the walls were ruled out as explanations for the observed behavior. The results are remarkable because all current theory suggests that such dilute polymer solutions should break up faster, not slower, than the analogous Newtonian case. Microfluidic uses of aqueous two phase systems include separation of labile biomolecules but have hitherto be limited because of the difficulty in making droplets. The results of this work teach how to design devices for biological microfluidic ATPS platforms.