Consider the situation in Can I register a Nebenwohnung before I have a Hauptwohnung?:
Suppose that married persons X and Y are moving to Germany. They will share a Hauptwohnung (primary residence) in city A, but X will additionally occupy a Nebenwohnung (secondary residence) in city B because X has a job at the other end of the country. So far so good: both will register as Hauptwohnung in city A, and X will additionally register as Nebenwohnung in city B.
I've heard that under certain circumstances, they might be able to deduct (some of the) costs for the Nebenwohnung that X occupies several days per week, from income tax — and perhaps the travel costs, too. For example, see Haupt- und Nebenwohnsitz: Rechtlichte Regelungen, which states, among other things:
Wird der Zweitswohnsitz nur aus berüflichen Gründen genutzt, besteht die Möglichkeit, die doppelte Haushaltsführung steuerlich geltend zu machen.
Wer steuerlich befreit werden will, muss mindestens 10% der Steuern, die auf den Hauptwohnsitz fallen, selbst bezahlen.
It's not clear to me what those conditions mean in practice. I'm not familiar with German legalese. For example:
When they say aus berüflichen Grunden, would that include that the second job is simply too far away, or does it mean for business purposes only? I read something about Betriebsausgabe vs. Werbungskosten but it's not clear to me if it can be deducted in either case or only if it counts as a Betriebsausgabe. In our case it wouldn't, because X and Y are both employees (both are Beambte, in case it matters).
And on the payment of the taxes for the Hauptwohnsitz. Are they talking about real estate taxes? Does that imply X and Y must own the Hauptwohnung in city A, or do tenants pay those taxes themselves too and would this condition not pose any problems?
It is useful to know this in advance, for it may be financially more advantageous to have a single shared primary residence together where only one person travels back and forth weekly, compared to a more symmetric situation where both travel back and forth fortnightly, as we did in the past in England.