When specifying phase and amplitude for a tidal constituent, what date/time does phase start from?
Tidal constituents follow the expression: $\eta(t) = A cos (\omega t + \phi)$, where $\eta$ is the tidal elevation at a specific time, $A$ is the amplitude of the constituent, $\omega$ is the speed of the constituent (the rate of change in phase), and, finally $\phi$ is the phase of the constituent at the initial time from which the time is measured. The phase, $\phi$, is measured with regard to the equilibrium tide.
Thus, the short answer is that the phase is usually referring to the delay (in degree) from the theoretical equilibrium tide. From the NOAA glossary for phase lag (or phase epoch, not to be confused with the tidal datum epoch):

Angular retardation of the maximum of a constituent of the observed tide (or tidal current) behind the corresponding maximum of the same constituent of the theoretical equilibrium tide. It may also be defined as the phase difference between a tidal constituent and its equilibrium argument. As referred to the local equilibrium argument, its symbol is $\kappa$. When referred to the corresponding Greenwich equilibrium argument, it is called the Greenwich epoch and is represented by $G$. A Greenwich epoch that has been modified to adjust to a particular time meridian for convenience in the prediction of tides is represented by $g$ or by $\kappa'$. The relations between these epochs may be expressed by the following formula:
$G = \kappa + pL$
$g = \kappa' = G – aS / 15$
in which $L$ is the longitude of the place and $S$ is the longitude of the time meridian, these being taken as positive for west longitude and negative for east longitude; $p$ is the number of constituent periods in the constituent day and is equal to 0 for all long-period constituents, 1 for diurnal constituents, 2 for semidiurnal constituents,and so forth; and a is the hourly speed of the constituent, all angular measurements being expressed in degrees.