The process of laminar mixing in a T-shaped micro-device is studied by direct numerical simulation for a model binary mixture, composed of two liquids having the same density and the same viscosity, yet presenting a strong fluidity of mixing effect, i.e. the viscosity of the mixture is a function of its composition. In particular, we consider the case where the viscosity of the mixture is up to three times larger than that of the pure liquids, with the maximum viscosity corresponding to either a 50 %-50 % (type 1 mixture), or a 25 %-75 % composition (type 2 mixture), to better emulate the behavior of real mixtures. The results are compared to the case with no fluidity of mixing effects (type 0 mixture), which has been largely investigated previously. In this latter case, the inlet streams remain separated up to a critical Reynolds number, corresponding to a strong increase of the degree of mixing. This transition is also characterized by a symmetry breaking, from a vortex flow regime, with a double mirror symmetry, to an engulfment flow regime, with a point central symmetry. When the fluid mixture has a larger viscosity than that of its pure components, a viscous layer forms at the confluence of the inlet flows, which tends to keep the two streams separated. Therefore, in this case, one would expect that the onset of the engulfment regime should be shifted to larger Reynolds numbers, in comparison with type 0 mixtures, with no sudden increase of the degree of mixing. Although this is what happens for symmetric, type 1 mixtures, for type 2 mixtures we unexpectedly find that, due to the lack of symmetry of the mixture rheology, the transition from vortex to engulfment regime, although occurring at larger Re, occurs suddenly, corresponding to a sharp increase of the degree of mixing.