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It's possible to send shares both to a localhost GBT "pool" (e.g., Bitcoin Core's) and to a remote Stratum pool.* Now, if I discover a block, won't both my localhost GBT "pool" and the remote pool claim it as theirs? This appears to be known as the "selfish mining attack."
*using, e.g., bfgminer's #allblocks option, which offers "Support for submitting found blocks to a local Bitcoin GBT server (bitcoind or Bitcoin-Qt with -server flag): just append #allblocks to the end of your bitcoind's URI." (source)