There are examples in real dimension $4$ of manifolds having two conformally equivalent Kahler metrics, which induce inducing the same conformal structure but with opposite orientation.

See the paper Ambikahler geometry, ambitoric surfaces and Einstein 4-orbifolds by Apostolov, Calderbank and Gauduchon.

There are examples in real dimension $4$ of manifolds having two conformally equivalent Kahler metrics, which induce the same conformal structure but with opposite orientation.

See the paper Ambikahler geometry, ambitoric surfaces and Einstein 4-orbifolds by Apostolov, Calderbank and Gauduchon.