Like some of the others above (kjd in post #9, for example), I was going to recommend MEK--Methyl Ethyl Ketone. I don't need to explain its particular advantages as it's already been done well enough above. I also do like his method of locating things temporarily with CA adhesives, and there are some super-thin versions of those, useful for their wicking properties.
The problem for me is that I probably can't get MEK now in Colorado, and I'm pretty sure you can't get it in California either (true of some other places too), but it might still be available here for certain uses, of which I have several. Not just for welding styrene to other styrene or even other plastics. It has some other uses with inks, and potentially ink printing.
Supposedly a bottle is on its way my way, and if it shows up, I'll be very careful with it. Else I'll have to find that old can from days gone by, which I might actually be able to do too. We'll see. But I'd rather not have to.
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Back in the day, John Nehrich, who made regular use of styrene solvents when making sub-shells for his well known brick structures used Solvent Weld (As I recall, SW was then, and still might be even now Testors liquid cement). He did point out in a video description of his method that you could "nudge it to get it just where you want it," so it did have at least some working time. Assuming that matters. It's been so long since I used MEK myself that I just can't remember if the working time is true for that solvent as well.
I'll bet kjd knows though....