Despite their popularity, convergence results of splitting methods have been largely limited to the convex/monotone setting. We present convergence results for several first-order methods, including the proximal point method, forward-backward-forward splitting, and Douglas-Rachford splitting, within a nonmonotone setting. To this end, we focus on a problem class characterized by an oblique weak Minty condition, which captures non-trivial structures as we demonstrate with examples. Moreover, we introduce the concept of semimonotonicity and provide sufficient conditions for the global convergence of splitting techniques for the sum of two semimonotone operators. Illustrative examples demonstrate the wide range of problems our theory is able to cover.