  We present the axiomatisation of the fuzzy bi-G\"{o}del modal logic
(formulated in the language containing $\triangle$ and treating the
coimplication as a defined connective) and establish its PSpace-completeness.
We also consider its paraconsistent relatives defined on fuzzy frames with two
valuations $e_1$ and $e_2$ standing for the support of truth and falsity,
respectively, and equipped with \emph{two fuzzy relations} $R^+$ and $R^-$ used
to determine supports of truth and falsity of modal formulas. We establish
embeddings of these paraconsistent logics into the fuzzy bi-G\"{o}del modal
logic and use them to prove their PSpace-completeness and obtain the
characterisation of definable frames.
