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It's not really about being anal regarding a realistic depiction of time travel but that rather in the absence of an explanation about specific mechanics regarding time travel in the OP world it's wiser to default to how these things would actually work out. That doesn't mean the OP world can't...

One Piece actually did it correctly. If you're only able to jump forward, the only way you can meet yourself in the future is if you're in an alternate timeline where you can meet a version of yourself that never time travelled. So assuming you're in the same time line, you simply cease to exist...