There is a poset constructed by combining in a certain way the usual order on the reals with any well-order on the reals (I can provide details if needed). I've heard it called the "Sierpinski Reals" and "The Leaning Tower of L'viv", but I do not see anything online when I search these names. Is this construction known by any other names?
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Edit: Also the term Sierpinski tower such as mentioned in Random Walks on a Fractal Solid.