We sketch a strategy to prove the Tate conjecture on algebraic cycles for a good amount of quaternionic Shimura varieties. A key point is a twisted adjoint L-value formula relative to each quaternion algebra $D/F$ for a totally real field $F$ and its scalar extension $B=D\otimes_F E$ for a totally real quadratic extension $E_{/F}$. The theta base-change lift $\mathcal{F}$ of a Hilbert modular form $f$ to $B^\times$ has period integral over the Shimura subvariety $Sh_D\subset Sh_B$ given by $L(1,Ad(f)\otimes(\frac{E/F}{}))\ne0$; so, $Sh_D$ gives rise to a non-trivial Tate cycle in $H^{2r}(Sh_B,\mathbb{Q}_l(r))$ for $r=\dim Sh_D=\dim Sh_B/2$.