or “unethically” (depending on your ethics!), you could just _say_ you have a competing offer. Unless there is like some secret hiring manager slack where people ask each other “did you really send that person an offer?”.
You really don't want to bring up a "competing offer" unless you're prepared to take the counteroffer/just walk away.
But say you don’t want to forge one for just any company because that might not be legal. You could make your own company with a site and make offer letters that way.
FWIW I've done this several times (multiple offer scenarios where I have companies compete against each other for me) and never once has a company asked to see actual proof of the other offers on condition of hiring me. If that was a condition, I would say, sorry, I'll go with one of the other ones