Is suchness, tathata, as a concept always something in addition to phenomena? Is suchness, tathata, as a concept always something in addition to phenomena?

If not, when we talk about the suchness of phenomena we could mean what they are like, the qualitative experience of a taste e.g..

But if, conceptually, they are different, then no experience of taste amounts to its "true state" -- the meaning of suchness.

I'm asking because I've seen it said that tathata means qualia. 

And, while I never agreed, I wasn't sure what's wrong with that. I suspect, today, that it's wrong because they are in fact opposites -- opposites that need to be harmonized
Your question is an interesting philosophical investigation, if you can break or crack a concept, it will bring you closer to enlightenment, at least. In fact the Gōng'àn (Jap: Koan) is doing the same thing but in the Eastern style, more intuition, and 觀.

I do not think that I can answer you with a word or few sentences to define what is tathata, or suchness. In fact no one has ever been able to do so, else why we left fumbling it but the Buddha not defined it definitely like he did to the aggregates, karma... or paramita?

But perhaps I can help to unknot the knot in your question. 


  Is suchness, tathata, as a concept always something in addition to phenomena?


No. Parted from phenomena there is no tathata existed independently, parted from tathata neither phenomena. It's like, you cannot say the ocean or a droplet existed independently parted from the water, and vice versa. 

Therefore you cannot say "something in addition to". Is the water in addition to the ocean that make the ocean an ocean? 


  If not, when we talk about the suchness of phenomena we could mean what they are like, the qualitative experience of a taste e.g..


I do not know where does this "suchness of phenomena" phrase come from? Perhaps reading some bad writings on Buddhism by Teacher XYZ? At least, in the Chinese text materials this phrase doesn't exist; this phrase construct is invalid and ludicrous. 

In Chinese Buddhism, tathata is 如, or 如如. The 
Western civilization doesn't have equivalent concept, but I try to bring it closer for contemplation. The English words if, as it is, such as, as... could be close; or thisness/thatness the textbook one but really a bit misleading though. Hence, while your "the qualitative experience of a taste" is just an "as it is", "if", "as"; how more you can say about an "as it is" for giving the qualitative favor?


  But if, conceptually, they are different, then no experience of taste amounts to its "true state" -- the meaning of suchness.


Again, you cannot put this 'no experience of taste amounts to its "true state"'; same as, is the true taste of the ocean the amount of all the taste of all the water, or just a drop of water? If it's the former, no existing being has ever known the taste of the ocean (perhaps it's sweet with caramel and orange tang? )


  Specifically, when Buddhists say phenomena are empty of own nature, do they always mean phenomena as such and not phenomena considered as themselves?


I can only say, tathata and svabhava are closely related but they are not the same. 

Again, you cannot say "phenomena as such and not phenomena considered as themselves", simply phenomena, or anything, cannot have two contradictory qualia(s). You cannot say the rat is in the mammal class also the mammal class is excluding rat. 

This last paragraph quoted is clouded with bubbling blurbs perhaps is unhelpful to understand the core teaching. Suggestion, drop it. Sometimes words and the mind's ability to construct syntax is a hindrance.  



To answer your real concern, 


  I'm asking because I've seen it said that tathata means qualia.


This saying is an erroneous error. Tathata does NOT mean qualia. If it is that simple to define, why the Buddha has to spend another 20-30 years to teach the 2nd and 3rd Wheel Turning Sutras?