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For this question, I think a good working definition of co-orbital configuration would be two bodies orbiting around a third much larger body in a 1:1 resonance and where neither mass is negligible. In other words, you can't really say that either one is at the other's $L_3$ because they are interacting with each other more as equals.

I found this article in the New Scientist in my first on-line search, but it has a retraction at the top.

Just for example, here is a theoretical analysis of stability in an ArXiv preprint on the subject: Dynamics of two planets in co-orbital motion.

With the caveats above (which I think are fairly standard for the term "co-orbital planet") have any been discovered?

edit: Have any new candidates been at least identified or proposed?

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One example is Kepler-91b; early observations included strange light curve behavior. While follow-up observations seem to confirm that it is in fact a planet, rather than some other object, Placek et al. (2015) suggested that there is a Trojan "planet" ($M\sim.025\pm.019M_J$) at one of Kepler-91b's Lagrange points - either L4 or L5.

The authors believe that this explanation explains the light curve data much better than the single-planet model, as the unexplained dimming could be the result of a "secondary transit" by the Trojan planet, rather than a "primary transit". However, predictions of the daytime temperature of the secondary planet fall squarely about 5,000 K, which is close to the temperature of the surface of the Sun. Albedo calculations are also higher than expected (0.49).

Therefore, the authors state that the light curve may therefore be a false positive, and Kepler-91b does not exist - and nor does the hypothetical Trojan planet.

Trojan co-orbital planets have been the main focus of theoretical and experimental research so far; see Giuppone (2012) for more information.

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  • $\begingroup$ Thanks for flagging this potential example of co-orbital planets. I think it still qualifies here because (at the time of your answer at least) I don't think it has been officially "subsequently retracted". $\endgroup$
    – uhoh
    Oct 17, 2016 at 13:22
  • $\begingroup$ I've only looked at the abstract. Does the paper indeed suggest Kepler-91b does not exist, or only that the proposed co-orbiting or Trojan planet does not exist? $\endgroup$
    – uhoh
    Oct 17, 2016 at 13:24
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    $\begingroup$ @uhoh It suggests that neither exists. $\endgroup$
    – HDE 226868 ♦
    Oct 17, 2016 at 13:25
  • $\begingroup$ OK, thanks for the answer and follow-up! I'll have a go at the paper in the morning. If you're so inclined, this question is really bugging me! Oh! You saw it already, OK! $\endgroup$
    – uhoh
    Oct 17, 2016 at 13:26
  • $\begingroup$ The statement about things at the CR3BP L3 being unstable applies to small things with negligible mass. If you put equal mass planets in circular orbits around a star for example, it may or may not be unstable by various definitions, but you can not invoke the "L-word" in that case (as I mentioned in the question). Sometimes people will refer to that location with the "L-word" without intending to invoke CR3BP assumptions, but without shared context it can be tricky. $\endgroup$
    – uhoh
    Oct 17, 2016 at 14:50

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