Can I insist a buyer match a competing offer even if I never prove it exists?

You can say it, but insisting on it without being willing or able to substantiate it tends to backfire more often than it works, especially with an experienced buyer or one advised by their own lawyer. Buyers generally understand that sellers may have other interest, but a claimed competing offer that's never described in any real detail, or that you're unwilling to have your broker or lawyer confirm even in general terms, is often treated as a negotiating tactic rather than genuine information, which can reduce a buyer's trust in everything else you tell them during the negotiation.

The nuance is that you're not obligated to disclose another buyer's identity or exact terms, and legitimate confidentiality reasons can explain some vagueness. The issue is specifically insisting a buyer match something you present as fact while giving them no way to assess whether it's real, which is a different thing from simply saying you have other interest and are weighing your options.

If you do have a genuine competing offer, working through your broker or lawyer to convey enough credible detail, without breaching any confidentiality you owe the other buyer, tends to be far more persuasive than an unsubstantiated demand. A business lawyer can help you navigate what you can appropriately share.