A combined numerical and asymptotic study is presented for a viscoelastic fluid, which behaves as a thixotropic yield stress fluid in the limit of large relaxation time. Homogeneous biaxial extension under a prescribed stress is addressed. The constitutive model is a viscoelastic partially extending strand convection model for the microstructure embedded in a Newtonian solvent. There are two important parameters: the ratio of yield stress to stress modulus and the ratio of retardation to relaxation times. The transient change that takes place if a configuration is stressed biaxially is investigated. Distinct asymptotic regimes are identified and governed by fast and slow time scales. Steady flow curves may be monotone or non-monotone, depending on the ratio of yield stress to stress modulus. In either case, a slow evolution of the apparent viscosity is found upon cessation of the applied stress. A regime where steady solutions exhibit extensional thickening in uniaxial extension, but not in biaxial extension, is consistent with prior observations. The von Mises criterion, which relates yield stresses in extension and shear, is retrieved for the special case of immediate yielding at low strain.