Tateinheit = Verletzung mehrerer Strafgesetze durch eine Handlung, z. B. Mord in Tateinheit mit Raub – That’s the definition my Duden Universalwörterbuch gives.
Muret Sanders says “in coincidence with” which is not particularly helpful.

I am not a lawyer, but I think the meaning here is that one action leads to several crimes being committed simultaneously.

In your case, the person in question must be accused not only of attempted murder/manslaughter but also of something else (perhaps mentioned under b) ??) – I dread to think what!

As to how to translate it … perhaps concurrent or concomitant come close?

I found 2 good google examples which use CONCURRENT CRIME:

In any case, under Texas law a convicted murderer is not subject to execution unless he is also guilty of a concurrent crime like rape or some other serious ...
www.newaus.com.au/news209bone.html

Art. 36. - If the criminal finally convicted is sued afterwards for a concurrent crime, the provisions of article 34 and 35 will apply. ...
www.era.int/domains/corpus-juris/public_pdf/ romania_criminal_code.pdf

hope it helps and a Happy New Year!