Wow, so many responses! We live in Queensland in Australia. If it helps with identifying things, we're up on the second floor of an apartment block, with our plants all in pots on the balcony. We have a bunch of different herbs, an avocado plant and three anaheim chilli plants, two of which are huge and producing chilliesÂ
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Just to clarify - there's two types of bugs that I've seen. There's the larger, whitish clear ones (still less than a mm big) that are on our two large chilli plants, which is where I found the fine strands in some of the upper branches, and are the subjects of my initial posts. They move really quickly when disturbed. I haven't been hunting them down recently, just keeping an eye on them, as they don't seem to be multiplying out of hand.
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The other ones, that are actually like moving specs of dust (really, really small, would not be able to get a video of them, could only see them when shining a torch on the leaves) I've found on the third, small, stunted chilli plant that I just uploaded photos of yesterday. These specs are obviously harming the plant. There's no strands on the stunted plant that I can see, and I can't imagine spider mites being that small (watched videos of them on youtube).
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If neem doesn't work for these things, what's the best option? Although the stunted plant is probably never going to produce fruit I'd still prefer to use something that's food safe as there's a bunch of edible things around itÂ