If you search for a guy by the name of Larry Smith, he has produced numerous rule sets for tri dimensional chess, although they appear to be directly reverse engineering games played on the Star Trek television series.
The official rules of the World Tri Dimensional Chess Federation start with the king and queen in the centre of the rear rank, flanked by two ...
If this discussion is about 打ち歩詰め, one hypothesis is
Betray and kill the king by the lowest rank piece is forbidden when Sengoku Period(Japanese warring period).
And the other consideration is
Since there are 18 pawns altogether from both players, using them for winning spoils the nature of game Shogi.
Here is a rule that I think works:
Every piece you capture can be dropped by your partner only after his/her next move.
This prevents the situation you mentioned. Of course, it is still possible to reach a situation where each team has the turn at one board but does not want to move, but the specific problem you ask about would be mostly eliminated.