We propose a deterministic global optimization algorithm for mixed-integer nonlinear bilevel problems (MINBP) by generalizing the Branch-and-Sandwich algorithm (Kleniati and Adjiman, 2014a). Advances include the removal of regularity assumptions and the extension of the algorithm to mixed-integer problems. The proposed algorithm can solve very general MINBP problems to global optimality, including problems with inner equality constraints that depend on the inner and outer variables. Inner lower and inner upper bounding problems are constructed to bound the inner optimal value function and provide constant-bound cuts for the outer bounding problems. To remove the need for regularity, we introduce a robust counterpart approach for the inner upper bounding problem. Branching is allowed on all variables without distinction by keeping track of refined partitions of the inner space for every refined subdomain of the outer space. Finite ɛ-convergence to the global solution is proved. The algorithm is applied successfully to 10 mixed-integer literature problems.