Even Galois representations "mod p"

To lift $\bar{\rho}$ to a geometric representation in the sense of Fontaine-Mazur, the standard technique requires that $\bar{\rho}$ is balanced, i.e. the dimension of a certain Selmer group must equal the dimension of its dual Selmer group (associated to a certain deformation problem). This is not the case for even representations. However, if one relaxes the $p$-adic Hodge theoretic condition at $p$ (crystalline or de Rham for example) then it becomes possible to sometimes lift an even $\bar{\rho}$ to a characteristic-zero representation $\rho:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{Z}_p)$ which is ramified at only finitely many primes as was demonstrated by Ramakrishna in "Deforming an Even Galois reprentation". In this paper, some even representations were lifted to characteristic zero for $p=3$. These lifts should not satisfy the de Rham condition at $p$. On the other hand, you're not expected to get geometric lifts even for small primes ($p>7$ was deduced in Calegari's paper).

If you're looking for lifts which satisfy a local condition at $p$ you will get lifts which are ramified at an infinite density zero set of primes. These may be constructed through an application of the lifting strategy of Khare, Larsen and Ramakrishna. You may be interested in knowing where such representations in general come from.

Also, it is worth noting that the standard geometric lifting technique does not apply for residual representations $\bar{\rho}:G_{K}\rightarrow \text{GL}_2(\mathbb{F}_{p^m})$ if $K$ is not totally real since in this case the associated Selmer and dual Selmer groups do not match up in dimension. This in no means implies that there are no geometric lifts when $K$ is imaginary quadratic (for instance).