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In a series of three papers, Monson and Schulte showed that, subject to some mild constraints, certain families of real reflection groups can be reduced modulo odd primes to yield finite string C-subgroups of orthogonal groups. Further, polytopes of any desired rank can be constructed this way. In this talk I will show that the latter is also true for orthogonal groups defined over any non-prime field of characteristic 2. Of course, the modular reduction method cannot be used for such groups, so the polytopes are constructed from scratch using analogues of reflections. This is a report on recent joint work with J.T. Ferrara and Dimitri Leemans.