I’m not on a computer where I can try Matlab, although maybe I can check it out later. Looking it over, it’s not clear to me where on that doc page it says that it will produce 2\times 2 blocks if the function is applied to a sparse indefinite matrix. Does Matlab do that when you try it? I thought Matlab used CHOLMOD, which pretty clearly says in CHOLMOD Docs that A needs to have well conditioned leading principal submatrices. That is a pretty strong hint that it isn’t pivoting for stability.
Update: I did try it on Matlab. It looks like they are using the algorithm MA57 which is in HSL. There appear to be an interface to HSL at HSL.jl. If it works as expected, it looks like you can do this and get back the separate factors to modify and enforce positive definiteness.