We investigate bi-Hermitian metrics on compact complex surfaces with odd
first Betti number producing new examples with connected anti-canonical divisor
using the general construction of \cite{abd15}. The result is a complete
classification for all \it unbranched \rm Kato surfaces and a classification up
to logarithmic deformation for \it intermediate \rm surfaces.Comment: Major revision: we found a gap in the proof of Theorem 5.1 and
withdrew it; a few Remarks adde
