- (She explained to me) is REPORTING SPEECH, and the rest {that(while her boy friend was driving her home...out of it) reported.
Though without the use of "that" the sense can well be imagined, we are very much inclined to retaining the same as the question is about narration change and that too with back shifting.
Barring exceptional exceptions, if the reporting speech is in past tense, the reported speech should be in the "corresponding past tense" of the tense used in direct speech.
Let us scrutinize the direct speech. It could be seen that except 'driving of the car' every thing else is shifted back. Alright. We would distance the "driving" farther by using Past Perfect Continuous Tense( had been driving)
"To have" can function as both principal and auxiliary verb. Not only in indirct speech but also in any demanding situation HAS HAD, HAVE HAD or HAD HAD is used all the white. We can still find in our mind's eye the milkmaid of Dr. Edward Jenner's description moving fearlessly among small pox patients because she HAD HAD Cow pox.
So all we can suggest is to put 'that', first , change ' was' to 'had been' next and retaining ' had had' last. The text is now perfectly balanced.
As regards use of "while", in whatever way we look at it — as a temporal preposition with time related complement clause or a conventional time subordinator(conjunction)for simultaneous activity/ reference to background to the focus-activity in the main clause, there is not much of a difference. We are to use past perfect in both as stated ealier.
6IMHO, there should be that before while. "She explained to me that while her boyfriend had been driving her home they had had a terrible argument." – CowperKettle – 2015-12-21T18:49:31.370
3@CopperKettle, you can omit that from the beginning of the reported speech. I also agree that was driving should be replaced by had been driving so that the direct speech proposed by the OP work. – Alejandro – 2015-12-21T19:19:29.810
1+1 for the "that before while". I personally don't know many native speakers who are so conscientious about tense shifting in reported speech that they'd say "had been driving her home..." – Tᴚoɯɐuo – 2015-12-21T19:20:12.320
@TRomano - you are right on the second count, as my foray into COCA attests. But coming to think of it logically, it's puzzling. There are two "pasts", and one is anterior to another. So how could one happen while another happens...
– CowperKettle – 2015-12-21T19:21:45.9373I'd omit the "that" if the verb phrase had been "She said, while her boyfriend..." but with "explained to me" I think your instincts to use "that" are correct. – Tᴚoɯɐuo – 2015-12-21T19:23:35.560
@TRomano, well that's true. I've seen some reporting structures and I've realised that they don't omit that unless the reporting verb is said. – Alejandro – 2015-12-21T19:30:11.690
https://en.wikipedia.org/wiki/James_while_John_had_had_had_had_had_had_had_had_had_had_had_a_better_effect_on_the_teacher Search this it might help you. – Sam Harrington – 2015-12-21T19:38:47.583
1We would also omit that if it had been, "As she explained to me, while...." In fact, we couldn't use "that" there. – Tᴚoɯɐuo – 2015-12-21T19:42:50.547
I'm curious what else you guys are going to come up with. The puzzle is more psychological than linguistic, I think. – Ricky – 2015-12-22T00:09:06.433
Ricky said the puzzle was more psychological than linguistic. – Tᴚoɯɐuo – 2015-12-22T01:14:42.370
Ricky said that the puzzle was more psychological than linguistic. :-) – Mark Hubbard – 2015-12-24T01:23:52.370
3Seriously now, if you omit "that" before "while" in the original poster's example, then it can be inferred you were present (i. e., sitting in the back seat of the car) "while her boyfriend was driving her home." And "had had" is correct as well in the original poster's example. I feel sorry for user5577. – Mark Hubbard – 2015-12-24T01:37:33.583