  We construct a commutative algebra A_x of difference operators in R^p,
depending on p+3 real parameters which is diagonalized by the multivariable
Racah polynomials R_p(n;x) considered by Tratnik [27]. It is shown that for
specific values of the variables x=(x_1,x_2,...,x_p) there is a hidden duality
between n and x. Analytic continuation allows us to construct another
commutative algebra A_n in the variables n=(n_1,n_2,...,n_p) which is also
diagonalized by R_p(n;x). Thus R_p(n;x) solve a multivariable discrete
bispectral problem in the sense of Duistermaat and Grunbaum [8]. Since a change
of the variables and the parameters in the Racah polynomials gives the
multivariable Wilson polynomials [26], this change of variables and parameters
in A_x and A_n leads to bispectral commutative algebras for the multivariable
Wilson polynomials.
