agree it's going to depend on the relationship between the agencies and how the 2nd one knew there had been an application at the first (they might share an ATS for example). And then it might depend on that authorization form.
Unfortunately the OP's spouse chose to write different answers on the two PHSs....it might also depend on what she disclosed vs left off. BG Checks for law enforcement/government positions are meant to weed out applicants with issues....

