Summary: Let \(d>0\) denote the discriminant of a real quadratic field. For all bicyclic biquadratic fields \(k=\mathbb Q(\sqrt{-3},\sqrt{d})\), having a 3-class group of type (3,3), the possibilities for the isomorphism type of the Galois group \(G= \mathrm{Gal}(k_3^{(2)}|k)\) of the second Hilbert 3-class field \(k_3^{(2)}\) of \(k\) are determined. For each coclass graph \(\mathcal G (3,r), r \geq 1,\) in the sense of Eick, Leedham-Green, Newman and O’Brien, the roots \(G\) of even branches of exactly one coclass tree and, in the case of even coclass \(cc(G)=r\), additionally their siblings of depth 1 and defect 1, turn out to be admissible. The principalization type \(\varkappa(k)\) of 3-classes of \(k\) in its four unramified cyclic cubic extensions \(K_1,\dots,K_4\) is given by \((0,0,0,0)\) for \(cc(G)=1\), and by \((0,0,4,3)\) for \(cc(G)\geq 2\). The theory is underpinned by an extensive numerical verification for all 930 fields \(k\) with values of \(d\) in the range \(0<d<5 \cdot 10^4\), which supports the assumption that all admissible vertices \(G\) will actually be realized as Galois groups \(\mathrm{Gal}(k_3^{(2)}|k)\) for certain fields \(k\), asymptotically.