There is not a phase in the usual sense in which there is a number $\phi_0$ such that
$$x_1(t)=x_2(t-\phi_0) \ \forall t$$
where $x_1(t)$ and $x_2(t)$ are two periodic signals of different frequencies.
However, there are some phase-like analysis one could perform, depending on the application. I'll talk about two examples that come across my mind quickly.
Suppose we have the two signals $x_1(t)=\sin(t)$ and $x_2(t)=\sin(3t-\phi_0)$. As the latter is the 3rd harmonic of the former, their sum is periodic. However, the shape of the resulting wave would vary a lot depending on $\phi_0$ (some kind of "phase difference" between the signals). In the image below, I plotted three different waves ($x_1(t)+x_2(t)$ for three different values of $\phi_0$):
- $\phi_0=0$ in red.
- $\phi_0=\pi$ in blue.
- $\phi_0=\pi/2$ in green.
As you can see, there is a strong dependence on the value of $\phi_0$. This would be very important in music, for example, as the resulting wave would have a pretty different sound depending on the "phase" between the harmonics.
Following the same example, there is another way to see the phase between these two signals. Take for example the case in which $\phi_0 = 0$:
As you can see, whenever the fundamental wave crosses the $t$-axis, its 3rd harmonic also does. On the other hand, whenever the first one reaches a maximum (a minimum), its 3rd harmonic reaches a minimum (a maximum). This could lead to one thinking that they are in anti-phase. Again, this depends on what your application is and what you are looking for doing this analysis, as well as on what you define as "phase".
As a sidenote, there is a typical example that uses the phase between two waves of different frequencies (or something like that). It's called PLL (or phase-locked loop). You can check that out if you are interested in the topic.