In my opinion one of the best examples of a physical statement depending on mathematically independent is provided by Malament-Hogarth machines (You can find a bunch of other links just by doing a google scholar search). These are machines which it is argued (though others argue they are impossible) are allowed by general relativity and allow the operator to determine the answer to any $\Pi^0_1$ claim in finite time. As, by Godel's theorem, given any computably axiomitizeable theory $T$ the $\Pi^0_1$ statement $Con(T)$ is independent from $T$. So whatever theory one chooses to work in this lets you produce a physical statement (the machine with such and such construction ... will give answer blah) that depends on the truth of an independent question.
Note that this doesn't specifically involve large cardinal assumptions themselves but it does give you a physical system whose outcome depends on the consistency of large cardinal statements (e.g. $Con(ZFC+exists measurable)$)
However, one should not assume that how these physical systems turn out tells us what is mathematically true. We develop physical theories by accepting those hypothesises that seem to be good descriptions of reality and it is usually convenient to write those theories in terms of the most natural mathematical structures like the natural numbers or the reals.
However, it is equally true that the theory which says experiments turn out the way General Relativity formulated in this non-standard model of the reals/integers predicts (almost certainly it can be formalized in the two-sorted first-order theory of second order arithmetic) is as compatible with all our evidence as the theory which applies General Relativity formulated in the standard model.
Ohh to put it more simply if your Malament-Hogarth machine tells you that a particular computation converges in finite time you can't really be sure that the computation really converges in finite time or if the temporal structure of the universe is non-standard and the computation only converges at some non-standard time.