I'm having trouble understanding, conceptually, why the rate of return for buying a bond, shorting the future, and then delivering the bond into the future is called the "implied repo rate". I hear that it has to do with essentially being a synthetic reverse repo, but I can't understand exactly what the parallel is to repos.
In a reverse repo transaction, one sells an asset and agrees to buy it back at an agreed upon price. This is in effect a securitized borrowing of money. The difference between the price the asset is sold, and the price that it is repurchased is the interest paid for borrowing the funds. The asset sold serves as the collateral or security for borrowing the funds. In the event the loan is not repaid, the lender has the asset and can sell it to recover the proceeds that is lent.
The lender is in a "repo" with the borrower, in that they bought the asset and agreed to sell it at an agreed upon price. This is similar to the "long basis" trade you describe, where one has bought a bond and short the future. The seller of the future has in essence agreed to sell the bond at the agreed upon "future" price. Hence, what you hear described as "synthetic reverse repo." The difference between these prices, less the interest earned from being long the bond, represents the interest the lender is earning from lending money to borrower and the rate is the "implied repo rate" that the short future is lending to the long future, as if they had entered into a repo trade.
With Treasury bond futures however, there are a number of bonds that can be delivered by the party that is short the future to the party that is long the future. For each of the bonds that are eligible to be delivered, the exchange determines a conversion factor where the future price is multiplied to make all the bonds equivalent to a 6% yield. As each of these bonds have different coupons and maturities, they will trade at different prices, and will have different "implied repo rates". The short future will naturally want to deliver the bond that is the "cheapest to deliver" (ctd bond). The ctd bond can change during the life of the contract depending on how interest rates move and the yield curve changes over the life of the contract. This optionality is referred to as the "switch option." This and other options that the long future has sold to the short future, in addition to interest rate moves, can impact the price of the future and "implied repo rate" over the life of the contract.
In a repo trade, a borrowing rate is agreed to and the price to rebuy is derived from this rate. In bond futures, the futures price is traded, and where the deliverable bonds are traded produces an "implied rate" on a loan.