Summation of electromagnetic waves gives a Poynting vector with E and B fields not perpendicular to each other I am interested in getting the overall intensity (which is defined as $I\equiv \,<S>$) with 2 or more beams of coherent and monochromatic plane waves at the same frequency. I understand that Poynting vector is not linear so I will have to construct the new $\vec{E}$ and $\vec{B}$ fields separately and then construct the summed Poynting vector. But this method has an issue: Once I add $\vec{E}$ and $\vec{B}$ fields independently, the summed $\vec{E}$ and $\vec{B}$ fields are no longer perpendicular to each other, and thus the overall intensity is reduced due to the $sin\theta $ carried in the cross product. Is this look correct? Should the new beam be an ordinary EM wave as well (if not then how to sum fields correctly)? Please help!
 A: If the beams are linearly polarized and have a phase shift of $\pi$ then they interfere destructively, so this makes perfect sense :) 

Proof like I promised in the comment:
Assume two plane waves with $\vec{E_1}\perp \vec{B_1} $ and $\vec{E_2}\perp \vec{B_2} $.
Lets look at the dot product of the sums:
$$(\vec{E_1}+\vec{E_2})\cdot (\vec{B_1}+\vec{B_2})=\vec{E_1}\cdot \vec{B_1} + \vec{E_1}\cdot \vec{B_2} + \vec{E_2}\cdot \vec{B_1} + \vec{E_2}\cdot \vec{B_2}$$
But the first and last terms are $0$ because the fields are perpendicular.
We remain with
$$\vec{E_1}\cdot \vec{B_2} + \vec{E_2} \cdot \vec{B_1}=E_1 B_2 \cos \theta_{E_1 B_2} + E_2B_1\cos \theta_{E_2 B_1} $$
Now, because $E=cB$ you can see that $E_1 B_2 = E_2 B_1 $. If you draw to your self the vectors you can prove quite easily that the angles add up to $180^\circ$ meaning they have the same cosines just with a different sign, so the whole product is $0$ and the sum of each field are perpendicular!
QED
A: I am writing this answer although it does not directly address the problem to clear up a few things:
Light beams are composed out of innumerable photons in confluence, and both, photons and emergent beams obey their respective Maxwell equations,  (quantized and classical respectively).
When paradoxes appear analyzing data classically it means that the quantum mechanical formulation is necessary to understand what is happening.

I am interested in getting the overall intensity (which is defined as$I\equiv \,<S>$ ) with 2 or more beams of coherent and monochromatic plane waves at the same frequency. 

This experiment has been carried out  with lasers  the split beam interfering  at the screen. 
So you can see in the video that the actual power falling on the screen can be from zero, to two blobs on each other, to interference patterns. So there is no conservation of energy at the screen. What is happening with conservation of energy? This video by the same professor demonstrates the quantum mechanical nature of light, where one has to count everything the source included in the quantum mechanical solution, which picks up the slack of energy conservation.
So the superposition of the two classical beams needs the knowledge of the quantum mechanical sources  to really follow the energy pattern ( the Poynting vector is an energy measure). The lazing phenomenon is par excellence a quantum mechanical phenomenon.
Now about adding E and B fields you are ignoring that the sinusoidal form that allows additions happens in the plane wave solution. Adding stuff no longer are a single plane wave. Either the plane waves cross at some plane, when you see the effect in the video above, or they have an angle and each goes through unaffected, unless a screen or measurement is done at the  crossover point. If they are completely superposed, they are the same beam, or if added by a beam splitter  back to the exact same direction the phase will give effects as in the video the interference patterns will subtract from the energy, so the phase between the two beams has to be back to 0.
I also want to emphasize that superposition is not interaction. Quantum mechanically it is the wave functions that are superposed and will show interference patterns; the interaction is what happens at the lazing source.
