We present detailed models of low- and intermediate-mass asymptotic giant branch (AGB) stars with and without the F-18(alpha,p)Ne-21 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 F-19 and Ne-21. The stellar yields increase by similar to 50 to up to a factor of 4.5 for F-19, and by factors of similar to 2 to 9.6 for Ne-21. While the F-18(alpha, p)Ne-21 reaction competes with O-18 production, the extra protons released are captured by O-18 to facilitate the O-18(p,alpha)N-15(alpha,gamma)F-19 chain. The higher abundances of F-19 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. Extramixing processes are proposed to help to solve this problem. Some evidence that the F-18(alpha, p)Ne-21 rate might be closer to its upper limit is provided by the fact that the higher calculated Ne-21/Ne-22 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 F-18(alpha, p)Ne-21 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.