Why don't two lamps interfere with each other? The reason is that they are not coherent. Each lamp is made of a large number of atomic wave sources that have random phase, the wavefront produced from each lamp changes with time so any interference effect will not be stable. To achieve interference the light from one lamp can be passed first through one slit then two slits. The single slit selects part of the wavefront and makes that the source, this source is then passed through the double slits to form two sources, each source originates from a single source so are coherent and form a stable interference pattern.
But, if the lamps are identical right down to the atomic level then they should form a stable interference pattern. Each lamp would be made of a large number of identical atoms giving out EM radiation with the same changing phase. If the starting phase of each changes at the same time then the superposition of the waves will not be altered, so the interference pattern will be stable.
I tried making an animation using Paul Falstad's ripple tank showing the interference between two sources. Each source is two atomic sources, when nothing changes the pattern is stable but if the phase of one of the atomic sources randomly changes the pattern changes. However if the phase of one of the atomic sources in each "lamp" changes randomly at the same time, the pattern is stable. Well that's what I tried to show but all the waves just made me dizzy.
One atomic source changing in 1 lamp:
One atomic source changing in both lamps at the same time:
Hmm, not very convincing, they both look like the peaks change, however in the second one it's more the brightness of the peaks that changes than their position. This could be an idea for an EE.