This paper addresses the strictly dissipative control problem for continuous-time nonlinear Markov jump systems via Takagi-Sugeno fuzzy rules. The modes of a devised fuzzy controller are assumed to run asynchronously with the modes of original system, which is widespread in practice and described through a hidden Markov model. A sufficient condition is acquired to ensure the stochastic stability and strict dissipativity of the closed-loop systems, based on which the design method of an asynchronous fuzzy controller is provided. Finally, one example is presented to illustrate the effectiveness and new features of the proposed design techniques.