The web of lies [have] obscured the true issue, which is the integrity of our system.
The web of lies [has] obscured the true issue, which is the integrity of our system.
The subject of the sentence is web, not lies. The verb has must agree in number with its singular subject web. The prepositional phrase of lies, which comes between the subject and verb, does not change the subject-verb relationship or make the subject plural.
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See Error Checklist for a compilation of 75 common writing errors.
For more exercises in identifying common errors,
take the M.B.A. exam and the Punctuation and Grammar Challenges.