  We present the necessary and sufficient conditions for constructing gradient
Ricci almost solitons that are realized as warped products. This will be done
by means of Bishop-O'Neill's formulas and a particular study of Riemannian
manifolds satisfying a Ricci-Hessian type equation. We prove existence results
and give an example of particular solutions of the PDEs that arise from our
construction. We also prove a rigidity result for a gradient Ricci soliton
Riemannian product in the class of gradient Ricci almost soliton warped
products under some natural geometric assumptions on the warping function.
