In this paper, we study simplicial hyperplane arrangements in real projective $3$-space. We give a necessary condition for the characteristic polynomial to have only real roots, valid also for non-simplicial arrangements. As application, we obtain combinatorial inequalities which are satisfied for arrangements with splitting polynomial. This allows us to prove that there are only finitely many different isomorphism classes of simply laced simplicial arrangements whose characteristic polynomials split over $\mathbb{R}$. We also provide an updated version of a catalogue published by Grünbaum and Shephard and review some conjectures of theirs.