The recently formulated mixed quantum classical rate theory (MQCLT) is implemented for a model system with two degrees of freedom. In MQCLT, one must compute the Wigner representation of the symmetrized thermal flux operator. This phase space flux distribution is then multiplied by the classical reaction probability to obtain the rate. The major computational difficulty is the multidimensional Fourier transform necessary for obtaining the Wigner distribution. The Fourier transform reintroduces a sign problem when attempting to estimate the MQCLT rate using Monte Carlo methods. Two different methods for overcoming the sign problem are explored in this paper. Numerical results are presented for a model problem of an Eckart barrier coupled bilinearly to a slow oscillator and compared with numerically exact results.