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Well, it'd be nice to be sure I've tried testing it but the difference isn't visible without any clothes on and it's too hard to test with clothes on because of random procs and stuff. I guess simcrafting it both way would be the right thing to do.
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If the stronger DoT is likely to get clipped (target will die soon, you refresh before leaving Eclipse, or you plan to use AC after the current Eclipse), you'd want to cast the stronger DoT first, giving it one or possibly two extra ticks before it is clipped.
If the weaker DoT is still ticking (particularly if it is Eclipe-boosted), cast the stronger DoT now. Don't overwrite a DoT with an Eclipse bonus with a DoT that has no such bonus (you could make an exception to this for a long chain of DoT spam while moving).
If for some reason, Lunar Shower won't matter for the second DoT (you won't be able to make two consecutive casts, or you already have LS*3 from casting at other targets), cast the stronger DoT first.
Otherwise, Zgollum is correct, you want to give the Lunar shower bonus to the stronger DoT. However the difference is fairly small (perhaps <0.3% total DPS).