  We study a class of Poisson tensors on a fibered manifold which are
compatible with the fiber bundle structure by the so-called almost coupling
condition. In the case of a $5$-dimensional orientable fibered manifolds with
$2$-dimensional bases, we describe a global behavior of almost coupling Poisson
tensors and their singularities by using a bigraded factorization of the Jacobi
identity. In particular, we present some unimodularity criteria and describe a
class of gauge type transformations preserving the almost coupling property.
