We present a multishift, multipole rational QZ iteration for block Hessenberg pencils in this talk. We show that its convergence behavior is governed by rational functions. The aggressive early deflation strategy is incorporated in the algorithm and we pay special attention to the numerical stability of the pole swaps. Numerical experiments exemplify the competitiveness and accuracy of the resulting methods. This is joint work with Raf Vandebril and Karl Meerbergen.