Implementing a filter with an LDI integrator why this phase jumps
What are the weird phase jumps?

Weird phase jumps, because you're using a time-varying system.  I suspect the little hiccoughs at \$\frac{F_s}{2} + nF_s\$ has to do specifically with the switched-cap nature of the filter; the big excursions at \$nF_s\$ are because the circuit is a periodically time-varying one.
If you felt up to an afternoon's worth of math you could look up someone's class notes on switched-cap filters, and grab someone's datasheet for the MF10 (TI's is good, but I can't get a direct link).  Then work out the actual time-domain equations -- they'll be periodic enough that at some point you can go to the z-domain, with decorations for what happens due to FET channel resistance, and due to the fact that the LDI output commences \$\frac{1}{2}\$ clock intervals after the input starts affecting the filter. (you'd have to make outright guesses about what the MF10's channel resistance and capacitances actually are).
I think that the little hiccough in phase at \$125\mathrm{Hz}\$ is either because the chip's clock phases aren't exactly lined up right, or because there's some lag between the actual chip's response and that of an ideal chip with no channel (or other) resistance.

Is it because of sampling?

I really hate the use of the term &quot;sampling&quot; for switched-capacitor filters, because the channel resistance of the switches is going to interact with the capacitances and make it not-quite-sampled.  Which is why I'm being careful to say &quot;periodically time varying&quot; rather than &quot;sampled&quot;.  I'm probably being too picky.
But yes, it's due to the periodically time varying, almost-sampled nature of the circuit.