  We study irreducible odd mod $p$ Galois representations $\bar{\rho} \colon
\mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_p)$, for $F$ a totally
real number field and $G$ a general reductive group. For $p \gg_{G, F} 0$, we
show that any $\bar{\rho}$ that lifts locally, and at places above $p$ to de
Rham and Hodge-Tate regular representations, has a geometric $p$-adic lift. We
also prove non-geometric lifting results without any oddness assumption.
