  The basis space in the triaxial projected shell model (TPSM) approach is
generalized for odd-odd nuclei to include two-neutron and two-proton
configurations on the basic one-neutron coupled to one-proton quasiparticle
state. The generalization allows to investigate odd-odd nuclei beyond the band
crossing region and as a first application of this development, high-spin band
structures recently observed in odd-odd $^{194-200}$Tl isotopes are
investigated. In some of these isotopes, the doublet band structures observed
after the band crossing have been conjectured to arise from the spontaneous
breaking of the chiral symmetry. The driving configuration of the chiral
symmetry in these odd-odd isotopes is one-proton and three-neutrons rather than
the basic one-proton and one-neutron as already observed in many other nuclei.
It is demonstrated using the TPSM approach that energy differences of the
doublet bands in $^{194}$Tl and $^{198}$Tl are, indeed, small. However, the
differences in the calculated transition probabilities are somewhat larger than
what is expected in the chiral symmetry limit. Experimental data on the
transition probabilities is needed to shed light on the chiral nature of the
doublet bands.
