Composite pulp compositions for making paper are disclosed that include agricultural residue fibers, softwood fibers, and water. The composite pulp compositions are made by blending an agricultural residue with softwood chips and simultaneously and mechanically pulping the mixture of agricultural residue and softwood in one or more TMP and/or RMP operations. The agricultural residue is present in an amount in a range from about 1% to about 30% of the oven dry weight of the composite pulp composition. Co-pulping the agricultural residue with the softwood improves the tear strength properties of the composite pulp composition. Blended pulp compositions are also disclosed that include a chemical pulp or chemi-mechanical pulp blended with the composite pulp compositions having agricultural residue fibers and softwood fibers. Paper products made from the composite pulp compositions and blended pulp compositions are also disclosed herein.