The rational numbers $\mathbb{Q}$ are central to number theory, so I think it would be reasonable to claim a connection between number theory and ``real” physics if there were a physical system with properties that can be measured experimentally and which exhibits special behavior when a physical parameter taking values in $\mathbb{R}$ takes on rational values.
There is such a system consisting of electrons confined to move in two spatial dimensions, subject to a periodic potential and in a magnetic field which is transverse to the two-dimensional plane of motion. This system was analyzed in work by D. Hofstadter and others and the plot of the energy spectrum of this quantum mechanical system is often called Hofstadter’s butterfly. The physics is governed by the ratio of the flux of the applied magnetic field through a unit cell of the lattice to magnetic flux quantum $\Phi_0=h/e$ and when this ratio takes on rational values the energy spectrum has a band structure determined by the denominator of this rational number.
A picture of Hofstadter’s butterfly can be found on the Wikipedia page
https://en.wikipedia.org/wiki/Hofstadter%27s_butterfly
and is attached below.
Experimental confirmation of this structure was found by two different groups in 2013 in graphene devices on hexagonal nitride substrates. Here the effect is slightly more complicated than described above and involves the relation of the applied magnetic field to the Moire pattern coming from the orientation of the graphene lattice to the boron nitride lattice. References to the experimental results can also be found in the Wikipedia page.