We present the discovery of the radio afterglow of the short gamma-ray burst (GRB) 210726A, localized to a galaxy at a photometric redshift of z ∼ 2.4. While radio observations commenced ≲1 day after the burst, no radio emission was detected until ∼11 days. The radio afterglow subsequently brightened by a factor of ∼3 in the span of a week, followed by a rapid decay (a "radio flare"). We find that a forward shock afterglow model cannot self-consistently describe the multiwavelength X-ray and radio data, and underpredicts the flux of the radio flare by a factor of ≈5. We find that the addition of substantial energy injection, which increases the isotropic kinetic energy of the burst by a factor of ≈4, or a reverse shock from a shell collision are viable solutions to match the broadband behavior. At z ∼ 2.4, GRB 210726A is among the highest-redshift short GRBs discovered to date, as well as the most luminous in radio and X-rays. Combining and comparing all previous radio afterglow observations of short GRBs, we find that the majority of published radio searches conclude by ≲10 days after the burst, potentially missing these late-rising, luminous radio afterglows.