  The Gasper and Rahman multivariate $(-q)$-Racah polynomials appear as
connection coefficients between bases diagonalizing different abelian
subalgebras of the recently defined higher rank $q$-Bannai-Ito algebra
$\mathcal{A}_n^q$. Lifting the action of the algebra to the connection
coefficients, we find a realization of $\mathcal{A}_n^q$ by means of difference
operators. This provides an algebraic interpretation for the bispectrality of
the multivariate $(-q)$-Racah polynomials, as was established in [Iliev, Trans.
Amer. Math. Soc. 363 (3) (2011), 1577-1598].
  Furthermore, we extend the Bannai-Ito orthogonal polynomials to multiple
variables and use these to express the connection coefficients for the $q = 1$
higher rank Bannai-Ito algebra $\mathcal{A}_n$, thereby proving a conjecture
from [De Bie et al., Adv. Math. 303 (2016), 390-414]. We derive the
orthogonality relation of these multivariate Bannai-Ito polynomials and provide
a discrete realization for $\mathcal{A}_n$.
