Without a tape, a fragmented market would have no common record. Every execution — lit, dark, internalised, auction — must be reported to it, so the tape is the authoritative history of what traded where and when.
It carries more than price and size. Each print has trade-condition-codes identifying auctions, late reports, odd lots, derivatively priced trades and out-of-sequence corrections, which is why naive volume sums are often wrong.
Example: a stock shows 8.4 million shares on the tape. Three million of that is off-exchange reported through a trade-reporting-facility, 1.1 million is the closing cross, and 200,000 carries an out-of-sequence code. Treating all 8.4 million as continuous lit volume would badly misread the day.
Related: tape-print, trade-condition-codes, securities-information-processor, time-and-sales