We introduce a~paraconsistent modal logic $\KGsquare$, based on Gödel logic with coimplication (bi-Gödel logic) expanded with a De Morgan negation $\neg$. We use the logic to formalise reasoning with graded, incomplete and inconsistent information. Semantics of $\KGsquare$ is two-dimensional: we interpret $\KGsquare$ on crisp frames with two valuations $v_1$ and $v_2$, connected via $\neg$, that assign to each formula two values from the real-valued interval $[0,1]$. The first (resp., second) valuation encodes the positive (resp., negative) information the state gives to a~statement. We obtain that $\KGsquare$ is strictly more expressive than the classical modal logic $\mathbf{K}$ by proving that finitely branching frames are definable and by establishing a faithful embedding of $\mathbf{K}$ into $\KGsquare$. We also construct a~constraint tableau calculus for $\KGsquare$ over finitely branching frames, establish its decidability and provide a~complexity evaluation.