Unfortunately this idea gets passed around alot, but it is based on some fundamental misunderstandings about photosynthesis.
The principal misconception here is that photoperiod is an important factor an attaining saturating rates of photosynthesis. This arises from extrapolating real-world data without understanding the physiological mechanisms underlying the process.
Light is absorbed first by the pigment complexes around photosystem II. The energy is transferred to the reaction center, H20 is split to O2 and H+ and electrons are transferred to photosystem I with plastoquinone. At PSI the electrons and H+ are used to reduce NADP+ to NADPH which is further used for a variety of cellular processes, such as fixing CO2.
There are two important rate-limiting steps here: 1) the capture of light energy, and 2) the transfer of electrons from PSII to PSI. The first step depends on light intensity while the second depends on the size of the plastoquinone pool.
The absoprtion of light takes on the order of 10^-12 seconds. For our purposes, that is instantaneous. The transfer of electrons from PSII to PSI is slower, but still on the order of fractions of a second, typically.
In other words, beyond fractions of a second, photoperiod is simply not an important factor in determining the rate of photosynthesis. The driving factors are: 1) the light intensity, and 2) the plastoquinone pool. This is true of instantaneous rates of photosynthesis. Over longer time scales organisms do adapt to ambient conditions and thus change their intrinsic rates of photosynthesis under given conditions, but that is beyond the present consideration. Besides, this adaptation is based almost entirely on ambient light intensity.
So if photoperiod, beyond a few seconds, is not an important driver in the rate of photosynthesis, where does the idea come from?
Corals in nature do indeed tend to experience increasing rates of photosynthesis after sunrise (as does everything) finally reaching saturated rates of photosynthesis typically after several hours. The important factor here is not the photoperiod, but rather the changing intensity. When the sun rises the light intensity is initially dim. As the sun climbs higher in the sky the intensity rises and the rate of photosynthesis rises until, eventually, the light intensity is high enough that it is saturating to photosynthesis. Photoperiod simpy has nothing to do with it and light intensity has everything to do with it.
For example, if you were to cover a coral in a dark box all morning and uncover the coral at noon it would take on the order of seconds, at most, to reach a saturated rate of photosynthesis, not hours. The light intensity is plenty to saturate photosynthesis. It doesn't matter if the coral has been exposed to light for many hours or a few seconds: if the light is bright enough, it is bright enough.
When we turn on the lights over our corals, assuming our light sources produce light bright enough to saturate photosynthesis (metal halides and T5's are definitely bright enough under most circumstances) the corals become saturated with light essentially instantaneously and the rate of photosynthesis remains at a similar rate for as long as the light is on (assuming some other factor does not become limiting).
Thus, 6 hrs on, 6 hrs off, 6 hrs on, 6 hrs off, won't lead to any significant difference in net primary production as compared to 12 on, 12 off or whatever combination you like, as long as there are 12 hrs of light. There simply isn't a good reason to think that the two should be qualitatively any different in their effects.
The only way in which I can conceive that a person
might see a difference is if the corals are experiencing photoinhibition. If the light source is too bright it can cause damage to the corals due to the build-up of reactive oxygen species and molecular O2 in the tissues. In that case splitting up the photoperiod might allow the coral to dissipate some of these nasty species and recover after a light period. Thus, 6:6:6:6 on

ff

n

ff could end up having different effects than 12:12 on

ff. We might see something similar with insufficient water flow, or food, or etc.
I would suggest here that the way to fix the problems is not to blast the corals with too much light or given them too little water flow, food, etc.
For practical purposes, no, this will not work. The very premises the idea is based on are flawed.