Concerning the doctrine of irresistible grace, Calvinists stress that the point of this doctrine is not that people will be saved against their will, but that the elect will come when called, so to speak. And if we grant the U in TULIP, this naturally follows.
Under our rubric of primary causality, though, the I becomes a collapsed, common-sense modification much as L did (link 1 below). The elect will indeed come when called. But we have argued for a doctrine of prevenient grace under which the non-elect get called, but don't come because they choose not to. And this would indeed be disagreed upon by Calvinists.
But why? In our item on unconditional election (link 2 below) we quoted Sproul as saying [Spr.CG, 33] that "if grace is obligated it no longer becomes grace. The very essence of grace is that it is undeserved." I asked, in that light, for an explanation of how receiving grace somehow equates with "deserving" it, and how making a decision qualifies as a "work."
Some enterprising Calvinist may at some point offer an answer. But if they do, I have some social data for them that they should consider before answering.
Our commentary on irresistible grace is derived from what you might suppose to be unlikely sources -- David deSilva's Honor, Patronage, Kinship and Purity and Zeba Crook's Reconceptualising Conversion. deSilva and Crook show quite clearly that the relationship between God and men is described in the NT in terms of an ancient client-patron relationship. Crook in particular explores the terminology used by the NT and compares it to the language of patronage in secular and Jewish documents, showing a strong match. (Hence, note that although "client" and "patron" are terms used most often of Roman reciprocal relationships, the basic elements, such as reciprocity, are essentially the same.)
What patronage means, very simply, is that rich people give gifts and favor to the poor, and the poor were expected to be loyal (faithful) in return. God is the rich one (hence phrases referring to the "riches of his glory" [Rom. 9:23] have more meaning than we realize) and we are the poor folk.
Readers may find more details on this social system in deSilva's and Crook's books, but for us the key is that the specific term at issue, grace, carried within the context of the client-patron relationship a certain meaning that is antithetical to Calvinist doctrine.
Consider these points [deSilva, 104ff]. The word grace was used "to refer to the willingness of a patron to grant some benefit to another person or group." Aristotle defined grace as "helpfulness toward someone in need, not in return for anything, nor for the advantage of the helper himself, but for that of the person helped."
So far, nothing unusual. Grace, all agree, is God's free gift. Crook agrees, adding that [C138] "grace" means not merely kindness, but the bestowal of some gift, though Crook prefers the more specific words "favor" or "benefaction" to emphasize the concrete nature of the "gift".
Not surprisingly, "salvation" is one of the benefaction associated with the Greco-Roman deities, although of course the means was not the same as Christianity offered -- 139, 148.
But there is more. "Grace" can also be used "of the response to a benefactor and his or her gifts, namely, 'gratitude'..." And this reveals a key point: one of the chief moral values of this day was that "grace must be met with grace; favor must always give birth to favor; gift must always be met with gratitude" -- even if it was only an acknowledgement that a gift had been bestowed [C136].
What this shows us is that, first of all and on a different topic, that of the relation of faith and works>, our good behavior is an expected result of grace and not required for it. See link 3 below.
Second, related to our topic at hand, "there is no such thing as an isolated act of grace. An act of favor and its manifestation (the gift) initiate a circle dance in which the recipients of favor and its manifestations must 'return the favor,' that is, give again to the giver...To fail to return favor for favor is, in effect, to break off the dance and destroy the beauty of the gracious act." 
Finally: "Neglecting to return a kindness, forgetfulness of kindnesses already received in the past, and, most horrendous of all, repaying favor with insult or injury -- these were courses of action to be avoided by an honorable person at all costs."  As Crook further puts it, it is ingratitude [C136].
From these insights it is clear that the paradigm of prevenient grace fits much better what the ancients would have understood to be the nature of the relationship between God and man. God gives grace; man responds -- if favorably, more grace is bestowed; if unfavorably, less is received.
And therefore, Sproul's observation that "if grace is obligated it no longer becomes grace" becomes essentially of no relevance once we are beyond the first round of "gracing". The question of whether regeneration precedes faith would be answered, "Yes, it does, and faith is followed by more regeneration if accepted; then by more faith, and on it goes." And oddly enough, this is the picture we have always been given of sanctification in the life of the believer. Grace enacted creates obligation and initiates a relationship of mutual obligation.
And there is more yet. The word "faith" in client-patron contexts  referred both to the dependability of the patron to do what he was entrusted to do, and the trust placed by the client in the patron. (As Crook notes, some form of fidelity was expected of BOTH sides of a patronal relationship , though how it was expressed in each case was different.)
In this light, the familiar passage in Eph. 2:8-9 -- "For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: Not of works, lest any man should boast." -- takes on a meaning that is not amenable to Calvinism. A key question is what "that" refers to -- what is the gift of God? Just grace? Or grace and faith? Calvinists conclude that "that" refers to both items, grace and faith, and there is nothing wrong with that grammatically (it is one option, not the only one), but in terms of the client-patron relationship, it simply doesn't wash. A patron gave a client grace; the patron did not give the client faith. Faith was the client's response to the patron's grace -- or, it referred to the "fidelity" and trust held by the client in his patrons.
Thus Eph. 2:8-9 contextually cannot support the Calvinist position, unless we assume that Paul used these words in a way that would have made absolutely no sense to his readers. The "faith" is either our response, or else, if it is a gift of God, it means it is His "faith" in us -- or rather, using the word as the ancients would use it, it is the gift of fidelity God has given, His own fidelity as a patron in saving us as He has promised.
The problem is that commentators on both sides view "faith" in terms of the modern defintion which includes cognitive assent. But that is not what is in view in the client-patron template.
One other verse pointed to in this regard is Phil. 1:29, which says "it is given in the behalf of Christ, not only to believe on him, but also to suffer for his sake..." "Believe" is the same Greek word used to refer to faith; hence faith is "given" or granted to us. But in the client-patron context, what would be granted to us is faith in the sense of depending on God, our patron.
This fits in precisely with the contextual admonitions telling us not to be afraid of adversaries (1:28) because God is our patron who saves us. It may be suggested that similar passages about faith (like 2 Peter 1:1 -- link 4 below) are also be better understood in the client-patron context.
Finally, Calvinist commentators who speak derisively of the suggestion that Arminian views permit a "grace that fails to do what it intended" are offering a misplaced sentiment. By design, grace in a client-patron relationship would never be subject to "pass or fail" because the success was in the very act of grace itself, regardless of who accepted or rejected it. Dishonor and shame was upon the one who rejected the favor, and nothing was taken from the giver as a result of the rejection. Their very graciousness was what brought them honor and glory (that is, public acknowledgement of worth and social value) -- which was not lessened or compromised by the negative reaction of ungrateful potential clients.
In conclusion, I think it is clear that we are doing the Scriptures a disservice when we allow writers of the 16th century, or even the 4th century, to determine for us what writers of the first century were thinking or saying. Neither Calvin nor Arminius, as far as may be seen, knew anything of Hebrew block logic or of client-patron relationships, which look to be essential keys to understanding important texts in this debate.