Let $K, F$ be number fields, and let $(\rho_\lambda)$ be a compatible system of $\lambda$-adic Galois representations $$\rho_\lambda:\mathrm{Gal}(\overline K/K)\to \mathrm{GL}_n(\overline{F}_\lambda),$$ where for each unramified prime $\mathfrak p$, the characteristic polynomial of $\mathrm{Frob}_\mathfrak p$ is in $F[X]$.

For each $\lambda$, we can define the endomorphism ring $$\mathrm{End}(\rho_\lambda):=\{\phi\in \mathrm M_n(\overline F_\lambda):\rho_\lambda(g)\circ\phi = \phi\circ\rho_\lambda(g)\quad\forall g\in \mathrm{Gal}(\overline K/K)\},$$ which is an $\overline F_\lambda$-algebra.

Is there any connection between the various $\mathrm{End}(\rho_\lambda)$? In particular, does there exist an $\overline F$-algebra (or even perhaps an $F$-algebra) $A$, such that $$\mathrm{End}(\rho_\lambda) = A \otimes \overline F_\lambda?$$

If $F =\mathbb Q$ and $(\rho_\ell)$ is the compatible system of Galois representations attached to an elliptic curve $E$ over $K$, then by Faltings' theorem, we can take $A = \mathrm{End}_K(E)\otimes \mathbb Q$. Can we say anything more for a general Galois representation, or perhaps for a Galois representation which is automorphic?