I had a very similar experience, engaging an anomalous aircraft as an officer in the Navy, which you can read about here:
Not so Fast: Anatomy of a Skeptic Hack Job.
Some juicy probative questions... :):
1. Was the targeting engagement feed of any form of Standard SM-2/3, F/A-18 or other semi-active fire control system used to ascertain the contact as well? and further then confirm back as a solid ping? Or was the painting intermittent?
1a. If you painted the contact, did it react in any form to that specialized EM energy?
2. If so, was the AN/SPY-1 air search array in agreement with this targeting feed? And was the datum and vector of the contact in agreement between systems?
3. I presume that the contact did not squawk any form of IFF code.
4. Did the contact 'enter' your air warning engagement envelope, or simply 'appear' inside it and light off the AAWC and AIC consoles?
5. How did the TAO or Air Boss react to the presence of the datum? Did he use any sort of red phone to report the datum beyond just the scene commander?
6. Did your NTDS system maintain a recording of the engagement sequence, which you could roll back and review later - did they do this that you know of?
7. Was there pilot visual confirmation, coincident with the Link 11/14 sharing of the contact's datum between all platforms in the group?
8. Did the datum return any kind of signal flux in the EM return back to the AN/SPY-1, which you had to adjust the phased array for?
9. Did the datum react to the presence of CAP aircraft approaching? Was that reaction defensive or playful?
10. When you broke off engagement, and showed no longer any interest in the craft, did it appear to want to draw the group back into a cat and mouse game?
11. Did the pilot try any form of direct 121.5 or 122.2 MHz contact with the target?
12. Was there a specific junior officer on the ship who seemed to be assigned special interest documenting this incident(s)?
OK probably enough from me on this....