Modern avatar of Englert's solution? Bosonic solutions of eleven-dimensional supergravity were studied in the 1980s in the context of Kaluza-Klein supergravity.  The topic received renewed attention in the mid-to-late 1990s  as a result of the branes and duality paradigm and the AdS/CFT correspondence.
One of the earliest solutions of eleven-dimensional supergravity is the maximally supersymmetric Freund--Rubin background with geometry $\mathrm{AdS}_4 \times S^7$ and 4-form flux proportional to the volume form on $\mathrm{AdS}_4$.  The radii of curvatures of the two factors are furthermore in a ratio of 1:2. The modern avatar of this solution is as the near-horizon limit of a stack of M2 branes.
Shortly after the original Freund--Rubin solution was discovered, Englert discovered a deformation of this solution where one could turn on flux on the $S^7$; namely, singling out one of the Killing spinors of the solution, a suitable multiple of the 4-form one constructs by squaring the spinor can be added to the volume form in $\mathrm{AdS}_4$ and the resulting 4-form still obeys the supergravity field equations, albeit with a different relation between the radii of curvature of the two factors.  The flux breaks the $\mathrm{SO}(8)$ symmetry of the sphere to an $\mathrm{SO}(7)$ subgroup.
My question is whether the Englert solution has a modern avatar, perhaps as the near-horizon limit of some solution.
 A: If the 1998 paper by Castellani et al. below is right,

http://arxiv.org/abs/hep-th/9803039

then all such solutions including Englert's elegant solution – cited as [13] above – are dual to $G/H$ M$p$-branes which are described as solutions interpolating between flat space at infinity and some $G/H$ configuration near the horizon. Let me admit that I don't quite understand the paper – it seems to imply the existence of lots of new objects in ordinary decompactified M-theory.
Of course, as discussed above, the lack of supersymmetry reduces the probability that it makes sense to try to locate the object in any dual description. The stability of such non-SUSY solutions is only restored in some "large radii" limit and is completely lost in the opposite limit. Still, there are some moral examples where it makes sense to trace unstable objects to a dual description although their masses and tensions of course fail to be calculable from SUSY which isn't there. Various authors such as Gauntlett et al.

http://arxiv.org/abs/hep-th/0505207

have mentioned Englert's solution but discarded it because of the instability. You may also want to look at the octonion membrane by Duff et al.:

http://arxiv.org/abs/hep-th/9706124

It's probably not quite equivalent to Englert's solution but seems to be an object of the same dimension with the same algebraic niceties hidden in the core.
Some possibly very similar candidate dual CFTs (also citing Englert) are constructed by Fabbri et al.:

http://arxiv.org/abs/hep-th/9907219

Also, Englert's solution might have a weak $G_2$ holonomy

http://www.sciencedirect.com/science/article/pii/S0550321302000421

although this statement of mine could be complete nonsense: I don't have the access to the full paper. In 2008, Klebanov et al. studied dual CFTs to squashed spheres

http://arxiv.org/abs/0809.3773

that are claimed to be of Englert's type.
A: I wanted to point out this paper of Gauntlett-Sonner-Wiseman 
http://arxiv.org/abs/arXiv:0912.0512
which shows explicitly that the Englert and Pope-Warner solutions (on any Sasaki-Einstein manifold $X^7$) can be obtained via holographic RG flows from the Freund-Rubin $AdS_4\times X^7$ solution. As you pointed out the Englert solution is perturbatively unstable. In this paper
http://arxiv.org/abs/arXiv:1006.2546
we showed that the Pope-Warner solution on $S^7$ is also pertubatively unstable. This result was later generalized to the Pope-Warner solution on any Tri-Sasakian manifold in 
http://arxiv.org/abs/arXiv:1110.5327
Let me also point out that there are supersymmetric (1/4 BPS) warped $AdS_4$ solutions with flux which describe non-trival IR fixed points of the ABJM-BLG theory. These were discussed here in supergravity
http://arxiv.org/abs/hep-th/0107220
and here (and other references later) 
http://arxiv.org/abs/arXiv:0806.1519
in field theory.
Finally there are even less supersymmetric (1/8 BPS) $AdS_4$ solutions with internal flux which can again be interpreted as the IR fixed point of an RG flow from the ABJM theory
http://arxiv.org/abs/arXiv:0901.2736
http://arxiv.org/abs/arXiv:1010.4910
