The monodromy group of a translation surface M is the Lie group spanned by all symplectic matrices arising from the homological action of closed loops at M (inside its embodying orbit closure). In the presence of zero Lyapunov exponents, Filip showed that these groups are —up to compact factors and finite-index b groups— constrained at the level of Lie groups presentations: they are either special quaternionic orthogonal groups SO*(2n) for odd n, special unitary groups SU (p,q) for p superior q, or exterior powers of SU(p,1). Nevertheless, it does not follow from these constraints that every group in this list is realizable as the monodromy group of some translation surface. By the work of Avila–Matheus–Yoccoz, it is known that every SU (p,q) is realizable. Moreover, the work of Filip–Forni–Matheus shows an example that realizes SO *(6), which coincides with the second exterior power of SU( 3 , 1 ). In this talk, we show that the groups SO *( 2 n ) are realizable as monodromy groups for infinitely many odd values of n . To this end, we find concrete examples of square-tiled surfaces constructed as quaternionic covers —generalizing the construction by Filip–Forni–Matheus— having such groups as their monodromies (up to compact factors and finite-index subgroups). Our construction is valid for every n in the congruence class of 3 mod 8, up to a zero-density set.