"EM" waves without the E possible right? (thought-experiment)

Get ready for crazy thoughts :)

We're out in space with two dipoles, separated by a few meters. One transmits, one receives, it's working perfectly of course. What if one of the dipoles were inside of an aluminum box?!

The aluminum is thick enough to completely block the electric field from reaching out but there is still a magnetic field that should technically be able to penetrate. Perhaps eddy currents will cancel some of that (or a lot of it) but something tells me it should be possible to let some magnetic field out while containing the electric field. Once the M field gets to the receiving antenna, it will induce the current, which will of course create the E field again. The point is that in between the antennas only the M field is carrying the EM energy.
I'm sure the signal strength will be much less, I'm thinking about half.

The same thought experiment can be done with removing the M field, for example some type of ferrite tube around the dipole with only the ends sticking out.

What originally brought me to these thoughts is that when I look at all the different antennas, I can't imagine that the E and M fields are always equal. I actually think that some antennas have a stronger electric field while others have a stronger magnetic field. An electric field alone can induce current and a magnetic field alone can induce voltage. Maybe the pretty picture of the EM wave with its two completely balanced E and M components isn't really like that.

Of course at the transmitting antenna, you inherently have both fields available so it only makes sense to bring both of them out for full power, but is it possible to transmit with just one of those?

Better think about 1/1000000 (-120 dB) if the box is made perfectly hermetic. A box assembled from screwed aluminium sheets without RF gaskets will leak some energy and may provide -60 to -80 dB shielding.

It's actually daily work of engineers working with EMI control rather than a thought experiment.
That would be the case, if the antenna distance is small compared to wavelength, which you didn't specify. Generally E and M field are linked, a magnetic antenna generates an E field in its vicinity and vice versa. In so far the question title is somehow ignoring the nature of EM waves.

Pure magnetical antennas exist, e.g. small loop antennas with d << λ.