The sequence of uses tenses in complex or compound Sentence

Solution 1:

In

The head of the audit is so obtuse that he had no inkling that funds were being ingenuously diverted.

"that he had no inkling that funds were being ingenuously diverted." is a separate and subordinate clause describing the way in which he is obtuse. Subordinate clauses focus on describing or adding to the main clause. As such, there is no reason for them to "agree" with the tense in the main clause.

Compare:

I have bought an umbrella that will leak /I have bought an umbrella that leaks / I have bought an umbrella that has leaked.

Solution 2:

There's no grammatical mistake in the sentence. I would leave out one of the that complementizers, but it's a stylistic choice. The real problem isn't grammar; it's misperception of grammar.

The presenting question

• What is the sequence of tense usage in a complex or compound sentence?

presumes there is a "sequence of tenses" rule in English that can apply in any complex or compound sentence. That means all English sentences, in effect, since simple sentences are rare.

And what would a "sequence of tenses" rule look like? It would tell one what the tense of a verb in a complex or compound sentence should be, given (apparently) what the verb tense is in the first clause (or the main clause) of such a sentence.

Of course, there is no such grammar rule. Grammar doesn't work that way. Tense of clauses in English normally reflects either generic phenomena (Warm air rises) or natural time of events (It rained last night).

Two conjoined sentences can be of any tense, present or past, e.g,

• Bill left yesterday, but Mary is still here.
• Mary is still here, Bill left yesterday, and Joe never showed up.

So much for compound sentences. As for complex sentences, everything depends on the verb. Some verbs, like say, can take any tense in their object clauses

• Bill said that he was going to the store.
• Bill said that he went to the store.
• Bill said that he is going to the store.
• Bill said that he is tired of going to the store.

Others, like confess, necessarily entail a previously-occurring complement clause:

• Bill has confessed that he stole the money.

Infinitive and participial clauses have no tense, but they may implicate a particular time frame, especially if they're adverbial:

• Before reviewing the troops, the Duke had lunch. (report of events)
• Before reviewing the troops, the Duke has lunch. (generic report or scheduled events)

Adverbial tensed clauses may appear in the sentence in several orders, since adverbs are usually moveable:

• After/Before he reviewed the troops, the Duke had lunch.
• The Duke had lunch after/before he reviewed the troops.
• After/Before he reviews the troops, the Duke has lunch.
• The Duke has lunch after/before he reviews the troops.

In the presenting example sentence (lightly edited)

• The auditor is so obtuse [that he had no inkling [that funds were being diverted]].

there are three tensed clauses; the main clause is a so...that construction with obtuse as the maximizing variable, in the present tense, describing a personal characteristic (metaphoric for 'stupid'), and exemplified by reporting a past event. Note that a present event would be just as useful as an example of stupidity. Note also that someone described in the past as being stupid would likely still be described that way in the present.

The present tense of the main clause has no effect on the tense of its complement clauses; the speaker could just as easily produce a that clause in the present tense:

• The auditor is so obtuse [that he has no inkling [that funds are/were being diverted]].

Have an inkling means 'know', which is factive, so it must report past events or a generic truth. Either will do:

• The auditor is so obtuse [that he had no inkling [that funds are being diverted]].

• The auditor is so obtuse [that he had no inkling [that funds were being diverted]].

So, there is no "sequence of tenses" rule that can apply here. Use whatever tense you mean to use.