In the following questions, a sentence is given with a phrase highlighted in bold. The given phrase in bold may contain an error. The options following can replace the incorrect phrase to make the sentence grammatically correct and contextually meaningful. The correct phrase that is to be replaced will be your answer. If the sentence is correct as it is, then select 'No improvement required' as the answer.  Only after the federal investigators had meticulously scrutinized the corporation's offshore accounts , they realized the magnitude of the systematic embezzlement that had quietly drained the subsidiary's reserves.  

The sentence tests the advanced grammatical rule of inversion. When a sentence begins with a restrictive or negative adverbial phrase such as Only after, Only when, Hardly, Scarcely, or Rarely, the main clause must follow an inverted word order. This means the auxiliary verb must precede the subject (Auxiliary Verb + Subject + Main Verb).  The original phrase "they realized" uses standard subject-verb order, which is incorrect here.  Option (b) correctly inverts the subject and verb to "did they realize".  Option (c) is incorrect because "did" must be followed by the base form of the verb (V1), not the past tense "realized". Option (d) is incorrect because "had" must be followed by the past participle (V3), not the base form "realize".