  We show that any 4-manifold, after surgery on a curve, admits an achiral
Lefschetz fibration. In particular, we show that the connected sum of any
simply connected 4-manifold with a 2-sphere bundle over the 2-sphere will admit
an achiral Lefschetz fibration. We also show these surgered manifolds admit
near-symplectic structures and prove more generally that achiral Lefschetz
fibrations with sections have near-symplectic structures. As a corollary to our
proof we obtain an alternate proof of Gompf's result on the existence of
symplectic structures on Lefschetz pencils.
