This paper investigates the design of asynchronous switched nonlinear H∞ filters for a class of continuous-time nonlinear switched systems with mismatching switching laws and L2 norm bounded disturbances. In this context, the switched nonlinear system is modelled as a switched Takagi-Sugeno (T-S) model with nonlinear consequent parts, i.e. where unmeasured nonlinear terms are kept in the nonlinear consequent parts in order to circumvent the occurrence of unmeasured premise variables, which is usually faced in conventional T-S modelling without nonlinear consequent parts. In this framework, asynchronous switched T-S filters with unmeasured nonlinear consequent parts are proposed to estimate unmeasured and/or disturbed system’s outputs, even when the filter's switching law mismatches the switched nonlinear system's one, which can be in practice unknown or imprecisely measured. Based on a candidate multiple Lyapunov function, combined with a H∞ criterion, conditions are proposed in terms of Linear Matrix Inequalities (LMIs) for the design of the considered asynchronous switched T-S filters with unmeasured nonlinear consequent parts. Compared with previous related works, these conditions have the advantage of being dwell-time-independent and less conservative, thanks to the incremental quadratic constraints employed to deal with the unmeasured nonlinear consequent parts. Furthermore, acknowledging that T-S models are only representing nonlinear ones on subsets of their state space, an optimization procedure to estimate the filtering error's domain of attraction is developed. Two illustrative examples are considered to validate the proposed results. An academic one is presented to illustrate the improvements brought in terms of conservatism by the proposed switched T-S filter design methodology with regards to previous related studies. Then, a case study illustrates the effectiveness of this proposal from a switched nonlinear mass-spring system inspired from related literature.