  We further develop the paraconsistent G\"{o}del modal logic. In this paper,
we consider its version endowed with Kripke semantics on $[0,1]$-valued frames
with two fuzzy relations $R^+$ and $R^-$ (degrees of trust in assertions and
denials) and two valuations $v_1$ and $v_2$ (support of truth and support of
falsity) linked with a De Morgan negation $\neg$.
  We demonstrate that it \emph{does not} extend G\"{o}del modal logic and that
$\Box$ and $\lozenge$ are not interdefinable. We also show that several
important classes of frames are $\birelKGsquare$ definable (in particular,
crisp, mono-relational, and finitely branching). For $\birelKGsquare$ over
finitely branching frames, we create a sound and complete constraint tableaux
calculus and a decision procedure based upon it. Using the decision procedure
we show that $\birelKGsquare$ satisfiability and validity are in PSPACE.
