Does "phase" in physics always have to expressed in radians or degrees? According to this post
What is a phase of a wave and a phase difference?

The phase of the wave is the quantity inside the brackets of the sin-function, and it is an angle measured either in degrees or radians.
$\phi$ ("phi") is a "phase." It is a constant that tells at what value the sine function has when $t=0$ and $x=0$.

From this, I suspect that phase needs to be in an angle. However, for water waves we sometimes say "half a wavelength/cycle/period out of phase".
Is it still fine to quantify "phase difference" in units other than radians or degrees?
 A: Your use of 'phase' applies to two sinusoidal functions of the same frequency, or a single sinusoid with a constant term added to its argument. Then the units would be an angular measure. 
However the term is sometimes applied to other functions such as non sinusoidal pulse trains as in a "phase locked loop".  Then defining phase as an angle does not work--it could be in units of time. In general the term can be used to describe how far two things are out of alignment, or out of synchronism.  Then there are many types of units that might be used.
A: It is completely ok to use either radians or degrees or any other angular measure to measure the phase of a wave or any other regular oscillation. It is also ok to say things like "half a wavelength out of phase" because it is clear what is meant, but it would be better to say "half a cycle out of phase".
It would not be correct to say "the phase difference was so-and-so metres" and also it would be poor usage to say "the phase difference was half a wavelength". That last example would probably be interpreted in the right way by anyone who knows about waves, but strictly speaking the phase difference ought to be expressed either in some angular measure or as a fraction of a cycle. So you could say "the phase difference was half a cycle". It may seem a bit nit-picky to allow this latter statement and not the one refering to wavelength, but since your question was about correct usage it is appropriate to be nit-picky in this case.
The main point to note is that phase, in the context of any sort of regular oscillation, is a way of referring to how far through the oscillation something is.
