Ruling Couple

Most rulers have spouses and many have mistresses (or "misters?") as well. However, often only one of the couple exercises rule and the other is merely to decorate the palace, provide heirs, seal alliances and entertain the king.

A ruling couple, on the other hand, are equal or near equal partners, and may even be Happily Married. Rather then one ruling and one staying in the palace they jointly rule. The rulers will rely on each other as trusted counselors and they will be The Good King and The High Queen in one. Perhaps they will show this by receiving audiences on two thrones. Perhaps the consort will have a regular seat in the royal council and a vote. Perhaps even the two of them will discuss deep and labyrinthine affairs of state during matrimonial activities. And likely they will be addressed as partners and written down as such in the chronicles. If historians refer only to the reign of Alice or the reign of Bob then this trope might be averted. If however historians regularly refer to the reign of "Alice and Bob" then it is a ruling couple.

This usually refers to a monarchial government, as monarchy is intentionally a Family Business. Only rarely can it be pictured in The Republic.

On many occasions, they will also be a Battle Couple.

Examples of Ruling Couple include:

Literature

Belisarius Series: Justinian and Theodora (and in Real Life) for the Romans; Rao and Shakuntala for the Marathans; Kungas and Irene for the Kushans.
Belgarion and Ce'Nedra of Riva, and Korodullin and Mayaserana of Arendia from the Belgariad.
- Rhodar and Porrenn of Drasnia divide their power: One for internal affairs, the other for external ones.
Selenay and Daren of Valdemar. In fact, any royal spouse will become a co-consort provided that he/she is also chosen as a herald. There are several examples of this in the back story.
Dune: Leto and Jessica. Leto is the senior in this case but Jessica has at least as much influence as any of Leto's councilors, is a central character and definitely does not Stay in the Kitchen.
Deryni: This seems to be Kelson's plan for Araxie in King Kelson's Bride, as it had been for Rothana in The Quest for Saint Camber. Since things with Rothana didn't pan out, and since Araxie is also a Haldane by birth, Kelson suggests triggering the Haldane potential in her as it has already been done in himself. Araxie is not averse to the idea and enters into thoughtful speculation on the matter; no Haldane has ever been Queen of Gwynedd before, and it isn't known if a female Haldane could have her potential triggered (or even that she carries it).
Eugenides and Irene in The Queen's Thief series.
The Lord of the Rings: Celeborn and Galadriel, though Galadriel often seems to be the senior partner.
After the second book in The Sword of Truth series, the hero, Richard Rahl, ascends to his hereditary title as ruler of the D'Haran Empire. Two books later, he marries Kahlan Amnell, the Mother Confessor (basically, the ruler of a race of women that can permanently brainwash anyone just by touching them) and become, ostensibly, the most powerful husband and wife duo in the world. While Richard is undoubtedly the more active and usually the more dominant of the two, the two are practically equals, with people sworn to serve Richard also sworn to serve Kahlan and vice-versa. There are not very many moments in the story where the two are allows to rule as a partnership (almost every time they get together, they're torn apart or one is taken out of commission via Diabolus Ex Machina), but there are smatterings of it here and there.
Cayleb and Sharleyan in Safehold. There are married rulers in the books, but these two are the only case where they're co-rulers with equal authority. A number of their decisions after their marriage are made partly to emphasize this fact.
The Silmarillion: Manwe and Varda
1632: Mike Sterns and Rebecca, though of course, Grantsville/USE is not a monarchy. Still the same sort of interaction applies.
Vorkosigan Saga: Aral and Cordelia are Viceroy and Vicereine of Sergyar. Not bad for a culture where, just a few years earlier, a woman had to have a sex change operation to be allowed onto the Council of Counts.
A couple of instances in Wheel of Time. The Sea Folk have the Mistress of the Ships (always female political leader) and the Master of the Blades (always male millitary leader)- which takes precedence depends on whether the Sea Folk are currently at peace or war. Among the Aiel, male clan chiefs are the political and millitary leaders while female shaman-types called Wise Ones are the spiritual leaders- as it's not unknown for a clan chief to marry a Wise One, this trope sometimes results. Finally, in Tarabon the King handles external affairs and the Panarch (highest-ranked noblewoman) handles internal ones, though they're not neccessarily married (both are elected independantly from among the nobility following the death or downfall of their predeccessor).

Live Action TV

Sheridan and Delenn on Babylon 5. In this case, it is a bit complicated, as rather then being heads of government, they are heads of a trans-state La Résistance that includes some government leaders. Later they both lead an interstellar confederation.
King Arthur and Queen Guinevere as of the end of series four of Merlin.

Video Games

Graham and Valanice of King's Quest. It's implied that Alexander and Cassima will be this as well.
In Dragon Age Origins, it is possible to set up Queen Anora and Alistair as the Ruling Couple of Ferelden, but only if both agree to such marriage and the latter's personality has been hardened (if not hardened, he ends up as a puppet ruler more interested in partying than being useful).

Real Life

Ferdinand and Isabella of Spain
William and Mary of Great Britain.
Nicholas and Alexandra of Russia, at least after World War I broke out. After Nicholas took personal command of the army and went to the front, Alexandra ruled in Petrograd in his stead. It didn't work out so well...
Suleiman the Magnificent and his wife Roxelana who was an important advisor to him after being freed from his harem.