I get to drive across this crossroads several times a week:
Every approach has a stop line.
Yesterday I and two other cars approached at exactly the same time - who has priority?
Might be wrong but whoever is to the right I think.
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Why, the vehicle on the right, of course! 
Somewhat of a Mexican standoff.
I think that’s right, or whoever’s driving the artic 
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Stop, then give way to the Left… 
after that it’s a question of who is needing to change carriageways/turning through the middle line… that puts them behind someone who is not…
If I recall correctly…
Give Way to the Right is where roads join and where there is No White (Stop) Line to order any/either vehicle to Stop/Give Way… just a wide open void … aaargh…
often preceded by the giant X roadsign
Is there a Stop sign facing each approach?
to whom is this question addressed ??
According to Gendarmes, this sort of 4-way Stop… is designed to calm traffic and stop any one racing through… each person has to decide their correct way to drive… after they’ve stopped, of course…
Or the police car 
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Yes there is
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That’s it then… immediately calming things down… right down… stop!
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The stopping part I understand - it is just when multiple cars arrive at the same time (it was 3 yesterday) who goes first?
I’d always go the ‘give way to the right’ rule 
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I was given a tip for dealing with these. On approach, take careful observation of the other three junctions, and adjust your speed so that you arrive either before or after the other vehicles. Then, in general, it’s first come first served.
It also works at roundabouts. By adjusting your speed as you approach, you can usually flow onto the roundabout without having to stop. As my instructor told me, a static target is much more likely to be hit!
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It’s only Give Way to the Right when there are NO Signs/instructions…
The intersections being questioned in this thread all have Stop signs and solid white (Stop) lines… therefore NOT Priority to the right… or is it ???
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But in the absence of an alternative, with all four cars at the junction, surely the give way to right is a sensible option.
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The options depend on what each driver is wanting to do… it’s as per the Highway Code…
I was reading about this recently and, as I’ve said previously, it’s a calming measure… makes folk take a pause and decide who goes where… hopefully signals by each vehicle (as appropriate) will help.
Therein lies madness, as each car gives way to the one on its right, and nobody moves 
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4 way stops are common in S Africa and rural US where they work fine - I think the problem in France is that they’re not only much less common , don’t have the appropriate sign (number 4/3/2 below the give way sign) but also the really stupid priorité à droit now only applies in certain situations - I think it also applies on mini-roundabouts but everyone around here (12) ignores it. However locally, some older drivers (I’m only 71) still seem confused by roundabouts.