From the question title, my first inclination was towards notions of "puppeteer" or someone "pulling the strings" of power, but it seems you have rejected that. Here are two more suggestions for a politician who does not have a formal role but who retains power and influence.
Back-seat driver is far more general than the political context; it may be "used allusively for any person who intervenes with advice and instructions". It can have a derogatory sense similar to armchair general, i.e. that the person may lack relevant experience and is unqualified to make their suggestions. However, it can be applied to experienced, veteran politicians who are keen to make their influence felt.
"I shan't be pulling the levers but I shall be a very good back-seat driver."
(Margaret Thatcher, 27 November 1990, following her succession by John Major as British Prime Minister)
Alex Salmond was accused of being a “backseat driver” yesterday – by dictating SNP policy weeks after he quit as the party’s leader. The former first minister demanded Westminster hand Scotland all powers short of foreign affairs and defence. (Daily Record, 16 December 2014)
More akin to éminence grise is backroom operative. Here "backroom" is being used in the sense "marked by the exercise of inconspicuous control and maneuvering". Note that while éminence grise has a second sense of "an eldery ("grey-haired") personage who is renowned for past accomplishments, and now acts as an advisor rather than a principal actor", and even in the sense of a powerful decision-maker is, I think, generally applied to an elder statesman, a backroom operative might be a younger and lesser-known figure. It seems to span a range of meanings from a shrewd negotiator and political wheeler-dealer, to someone whose power stems from control of internal party procedures:
Silver is one of Albany’s most storied political figures, a consummate backroom operator with the power to single-handedly decide the fate of legislation. Along with the Senate majority leader and the governor, he plays a major role in creating state budgets, laws and policies in a system long criticized in Albany as “three men in a room.” He controls, for example, which lawmakers sit on which committees and decides whether a bill gets a vote. (Associated Press, 23 January 2015)
Mitch McConnell may soon be one of the most powerful people in America ... He is a formidable fund-raiser, strategist and backroom operator. He is unafraid of being disliked (he once referred to himself as Darth Vader, although he more closely resembles a retired librarian). (The Economist, 1 November 2014)
Inside the Brotherhood, Morsi has long been a backroom operator who dealt with security – often liaising discreetly with the Mubarak regime – and internal discipline. Little-known to the wider public, Morsi is a famously boring speaker who reduces Egyptian journalists to teeth-gnashing frustration as he rarely says anything remotely quotable. (The Guardian, 25 May 2012)
These examples all suggest political power greater than their formal position alone would indicate, that the exercise of their power is discreet or shadowy, and (as the question requires) they need not "look the part" of a traditional political frontman. I'm not sure that "éminence grise" would apply, except perhaps for Silver.