In 1987, Murty and Ramakrishnan proved the Tate conjecture for Hilbert modular surfaces. In this talk, we report on a joint project with Ambrus Pal to prove the Tate conjecture for equicharacteristic analogues, called Stuhler surfaces. They are moduli spaces of Frobenius-Hecke sheaves, introduced by Stuhler in 1986, and are a special case of stacks of G-shtukas. We will explain the similarities with Hilbert modular surfaces, and how to use p-adic cohomology to replace analytical tools. For example, a key ingredient in the proof is the semistable Lefschetz (1,1)-theorem of Lazda-Pal.