'Faith' - or more properly in-game, 'Religion' is a tricky concept to implement, because:
1. You absolutely cannot try to represent any actual religion or you will, inevitably, upset someone. And if you upset enough someones, your game is wrecked in the marketplace.
2. Religion historically was very much like Money: it could be a goal in and of itself, but more often had many uses and effects ranging from cultural to political to military. 'Faith Civ' can, therefore, mean almost anything, depending on what aspect the game design emphasizes.
3. Religious influence of any kind varies wildly. Early in the game, it is almost ubiquitous and pervasive: numerous Civs were all or in part defined by their religious practices and beliefs, even without 'theocratic' governments. Later, religion became less important during and after the Enlightenment and afterwards, but it never disappeared, and the amount by which it became 'less important' has been exaggerated: it was always of great importance to a large part of the population even if the Leaders of the Civ were no longer obviously religious. Recently 'religion' in various forms has apparently become more important as a cultural, political, diplomatic driving force, but it has not been remarked on (enough) that it frequently is important as an Alternative to secular institutions that appear to have failed to address the population's problems.ate Change, Wealth Inequity, and general constant change in culture and technology that seems to be too fast to keep track of for many people. (Apparent or Real) Threats abound, and religion has always been one answer to Threats of all kinds.
4. Religions Change. Although, especially those that have Holy Books defining them, religions try to remain absolutely firm and static, they never do. Therefore, there is always a dynamic between the 'Old Time Religion' and the New Religion, however each of those are defined - and just to keep things soap-slippery, the definitions keep getting changed even as members of the religion argue over them. The stresses of this religious dynamic have been sadly neglected in Civ, I suspect because it is too often assumed that this only becomes important when religions are afflicted with Major Schisms like the Protestant - Catholic or Sunni - Shia splits in, respectively, Christianity and Islam. However, I think that just as important is the constant reaction of religions to both outside religious influences and on-going secular changes in culture and technology that require response for the religion to remain relevant. The idea that you, the Great Poobah or Leader, can set all the aspects of the Religion for your Civ and then forget about it until you want to change it is, frankly, a fantasy model. In fact, your religion will keep changing whether you want it to or not, and the changes may be largely Unpredictable rather than Desired.
So, to return to the concept of a 'faith-based Civ', the whole concept, early in the game, is almost redundant: ALL Cis, to some extent, were 'faith-based' - even if their religion had very little influence on their political or military actions (like Classical Greece or Rome) it still did a lot to define the cultural aspirations of their populations. That influence could be enormously changed if a 'new' religion appeared, either because large parts of the population rejected it and a religious Civil War resulted (Cue the Protestant movement in Europe 16th - 17th centuries CE) or because it swept through the population and changed a great deal of the culture and even the organization of the Civ and population (Cue the late Roman Empire and the effects of Christianity). A sufficiently pervasive and coherent Religion could, ultimately, Define a Civ (Cue 'Arabia' or Renaissance Spain or Byzantium) - making it the ultimate 'Faith-Based' Civ.
And as noted, even modern, post-Enlightenment and secular Civs still have a large Faith Component, even if primarily confined to Culture, and a purely Theocratic or 'Faith-Based' Civ is still quite possible in the 21st century. In fact, if Ideology is defined as having many of the aspects of a Religion (Communism, Fascism, Libertarian Capitalism) - which is debatable but certainly possible and playable in a game design - then Faith Based Civs can be viable from beginning to end of the game.
