Coherence of waves - phase difference constant OR frequency the same? Wikipedia states,

"In physics, two wave sources are perfectly coherent if their frequency and waveform are identical and their phase difference is constant"

In an A Level Physics Marking Scheme, students are often asked to define coherence. The mark is scored for saying coherence is when the waves 'have a constant / fixed phase difference'
We ignore and give no marks for a comment that says 'the frequency / wavelength is the same'.
In my head, the frequency of two waves is the same if and only if there is a constant phase difference between the waves.
So does anyone have an idea why the mark scheme classes 'the frequency is the same' as insufficient?
 A: Imagine you are sending a pulse $p(t)=1, |t|<T_p/2$ if your information is bit $"1"$ and a pulse $p(t)=-1, |t|<T_p/2$, on a carrier $cos(\omega_c t)+\theta_0$, that is the transmitted RF pulse is $s(t)=p(t)cos(\omega_c t+\theta_0)$; this is called bi-phase modulation. The phase $\theta_0$ is some constant phase signifying some initial time known only to the transmitter.
When you receive the signal at some random location the delay in time is unknown but is assumed to be much less than the pulse length $T_p$ only the carrier phase changes by some fixed but unknown amount, that is the receiver sees $cos(\omega_c t+\theta_1$) but while $\theta_1$ is constant it is unknown to both the transmitter and receiver.
The receiver demodulates with an oscillator of the same frequency $\omega_c$ but with an unrelated phase, that is the receiver has a local oscillator of carrier $cos(\omega_c t+\theta_2)$, here $\theta_2$ is known to the receiver and is constant (for a good oscillator). By applying a so-called "quadrature mixer" the demodulation process multiplies the incoming signal $s(t)$ with both $cos(\omega_c t+\theta_2)$ and $sin(\omega_c t)+\theta_2$, and after filtering out (suppressing) the terms at double RF frequency $2\omega_c$ and keeping only the low-pass terms it results in two terms $r_c=p(t)cos(\psi)$ and $r_s=p(t)sin(\psi)$ where $\psi=\theta_1-\theta_2$ is assumed constant,
Obviously, if you knew what $\psi$ was you could tell from $r_c$ and $r_s$ what the sign of $p(t)$ is.
Even though $\psi$ is unknown to both the transmitter and the receiver it is still OK, because you could assume that, say, the very first pulse in every transmission be always positive, (a transmission protocol between the transmitter and receiver), that is representing bit $"1"$. So from values of $r_c$ and $r_s$ obtained for the first pulse you can estimate, i.e., measure, $\psi$ and as long as that does not change you can use it to derive every other bit. This is a phase coherent receiver.
A: After scouring the internet for the last day, I have found an answer hidden away on this very site. It is incredible.
This is by @Floris on this post:

But here is one way to look at it. Imagine a Young's slits experiment. You get a certain fringe pattern. Now put a piece of glass in front of one slit that is just thick enough to change the phase by 180 degrees for one slit only. Now the fringe pattern will shift - where there used to be minima you will have maxima, and vice versa.
If you alternately have the piece of glass there, and to there, the interference pattern will keep shifting back and forth - and you end up with no pattern at all. This is why a constant phase difference is important.

It clearly explains that subtle difference and why demanding a constant phase difference is imperative more-so than frequency in defining a coherent light source.
There is also a second answer which adds to the answer and gives further insight.
