Suppose f = (f1, …, fm) is a solution of a non-hyperbolic quasi-linear system of the form with fk = φk on ∂Ω, where each fi is a bounded function in C2(Ω) ∩ C0(Ω̄). For a system of equations that can have a slightly more general form than above, when Ω is an unbounded open subset of a slab SM and as |X| → ∞ in a specified manner and when certain other conditions are satisfied, a Phragmèn–Lindelöf theorem that yields the limits at infinity of the functions fk(X), k = 1, …, m, is proven.