Very few Calvinists are born the moment they give their lives to the Lord. The reason is there are so many hoops one must jump through, rationally, before one can arrive at a reasonable theology. And that's just getting person set up to wrap their minds around the very broad sovereignty of God.
Calvinism is a system, and I've never met an unbeliever who embraces a system. What they do embrace is the simple Gospel which tells them of God's love for them and of their sin, and of their need for Christ to redeem them.
Saving faith does not require passing a TULIP test, at least it's not spelled out that way in my Bible. Your mileage -- and your patience with me -- may vary.
I would have a difficult time with any Calvinist who would tell me otherwise, because so many Calvinists were first saved under Arminian pastors. Where would the world of Calvinism be today if not for the encouragement of free will Arminians who dared to hold an altar call? How many Calvinists would we have today if they were required to explain limited atonement before someone prayed with them to receive the grace and mercy of God?
This is not an argument for Arminianism. Not at all. However, there have been plenty of non-Calvinists such as Wesley whose earnest preaching of the simple Gospel should be accredited to them as great faith. You don't have to endorse their theology. You just have to admit that there is a benefit to having a few Arminians for the building of the Kingdom.
My pastor has a saying I think he borrowed from his seminar professor: "I don't care how much of a Calvinist you are, you're still going to preach the Gospel like you're an Arminian." The point is regardless of your theology, there's no alternative to the zeal of preaching the Gospel.
Just something to keep in mind the next time you unsheathe your sword to do battle with a semi-pelagian poopyhead.