We present detailed models of low and intermediate-mass asymptotic giant branch (AGB) stars with and without the 18F(a,p)21Ne reaction included in the nuclear network, where the rate for this reaction has been recently experimentally evaluated for the first time. The lower and recommended measured rates for this reaction produce negligible changes to the stellar yields, whereas the upper limit of the rate affects the production of 19F and 21Ne. The stellar yields increase by ~50% to up to a factor of 4.5 for 19F, and by factors of ~2 to 9.6 for 21Ne. While the 18}F(a,p)21Ne reaction competes with 18O production, the extra protons released are captured by 18O to facilitate the 18O(p,a)15N(a,g)19F chain. The higher abundances of 19F obtained using the upper limit of the rate helps to match the [F/O] ratios observed in AGB stars, but only for large C/O ratios. Extra-mixing processes are proposed to help to solve this problem. Some evidence that the 18F(a,p)21Ne rate might be closer to its upper limit is provided by the fact that the higher calculated 21Ne/22Ne ratios in the He intershell provide an explanation for the Ne isotopic composition of silicon-carbide grains from AGB stars. This needs to be confirmed by future experiments of the 18F(a,p)21Ne reaction rate. The availability of accurate fluorine yields from AGB stars will be fundamental for interpreting observations of this element in carbon-enhanced metal-poor stars.