I would guess that there will remain at least a few tatters of the Constitution left, and that ex post facto laws might get into trouble. The Constitutional rule reaffirming birthright citizenship doesn't actually do much except to prevent laws from being made that abridge it. I suspect that if a new law were written to limit it, there would be an immediate constitutional challenge, and we'd be back to the court. But in any case, I think any law that limits birthright citizenship would have to apply only to babies born after the enactment of the law.
e.t.a. Assuming the Thomases of this world do not figure out another way to deem the Constitution itself unconstitutional. When the most blatantly, flagrantly corrupt Justice is in thrall to the most egregiously, floridly corrupt President, and it seems nobody has the gumption to stand up to Long Dong Silver and his posse, nothing is off the table.