We consider bounded solutions of the semilinear heat equation $u_t=u_{xx}+f(u)$ on $\mathbb R$, where $f$ is of the unbalanced bistable type. We examine the $\omega$-limit sets of bounded solutions with respect to the locally uniform convergence. We show that even for solutions with initial data vanishing at $x=\pm\infty$, the  $\omega$-limit sets  may contain functions which are not steady states. Previously, such examples were known for balanced bistable nonlinearities. The novelty of the present result is that it applies to a robust class of nonlinearities. Our proof is based on an analysis of threshold solutions for ordered families of initial data whose limits at infinity are not necessarily zeros of $f$.