Paperdoc's response, as always, is good to think about on. I think that the OP still maintains some misconceptions.
First, two photons cannot just cancel each other out. True, you may have heard that the photon is its own antiparticle, but that does not come into these discussions of light and superposition. We still require energy conservation. As such, interference of waves can never destroy energy for a lossless medium. Instead, interference directs the energy to specific regions of space. So in the double slit problem, we see areas that experience an increase in energy density (constructive interference) and areas of an associated decrease (reconstructive). But when we talk about photons having a phase, the photons are not canceling each other out. The phase is associated with the wave function. The wave function is an equation that describes the probability density of the problem. By operating on the wave function with the suitable operator, we can get the statistical probability densities or averages of physical values like energy, momentum and spin. We can also get the probability density of the detection of the particle in a given volume. So the phase comes into play in how the full wave function behaves and it is the wave function that undergoes interference. Note that the particles are not necessarily interfering with each other. The actual physical meaning of the phase of the wave function is, if anything, a philosophical consideration that may or may not be defined by the interpretation of quantum physics. Most physicists are of the Copenhagen school and follow a shut up and calculate mentality when it comes to trying to physically define such qualities (the reason being is that while we can find analogies to classical physics, these properties are not experienced in our macroscopic viewpoint. You can interpret them in multiple ways and still end up with the same predictions). The double slit experiment can be performed using individual particles, one at a time, and still produce the same interference pattern that we would expect if we had a steady stream. That is, the effects of the interference can occur even when there are no other particles currently present to interact with (and with photons being bosons, they do not interact with each other very much).
Second, the ideas that you are attributing to photon behavior applies to classical theories of light. It just happens that the quantum behavior reflects very strongly the classical theory in many ways and people often mistakenly intermix the ideas. The phase of an electromagnetic wave only has meaning when we talk about a steady and large number of photons being emitted. The problem of why a random source emits an efficient power spectrum should be considered from the view of classical physics as the solution here would reflect the solution in the quantum theory. There is no need to consider photons, answering the question interns of the classical picture will answer the question in the quantum picture.
So looking at it purely as a classical wave, the only real question to answer is why does a random source allow for omnidirectional emission with a nonzero power spectrum? This again has to do with the fact that it is completely uncorrelated merely by the fact that it is random. Over time, the amplitude of a given frequency will shift as will the phase. This results in what is called temporal and spatial deconstruct. If there was an amount of correlation, or coherence, then there would exist some degree of consistent interference. But without this coherence, the waves that make up the spectrum are forever changing and thus cannot have a consistent behavior of interference. Note that this property of coherence is independent of the power of the signal. It matters not if we increase the power density, the incoherence of the random signal persists and thus is not a factor as you suggest.
Mathematically, we could state that at a given frequency, the wave will look like:
A(t)*exp{i\omega*t+\phi(t)}
where the amplitude A and phase \phi change randomly as time progresses.
I would suggest learning more about noise like uncoloured (white) and coloured (e.g. pink) noise. Other topics that would be relevant are correlation and coherence. As Paperdoc pointed out, these are general properties of signals and while it may be more often discussed in the context of an electrical or audible wave, the electromagnetic waves follow the exact same physics.
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