It’s kind of hard to square that power with the claim that the states are "co-sovereigns" with the federal government. It’s an odd kind of "co-sovereign" that exists by grace of its "peer."

Congress has never voided a state government under the clause, and probably never will. The wording, however, doesn’t specifically refer to Congress, but to the entire "federal government." The 10th Circuit Court of Appeals decided that a group of Colorado legislators could bring suit against their own government, on the grounds that it is not "republican" in form.

I’ve written before (here and here) about the case, Kerr v. Hickenlooper. It’s an attack on a provision of the Colorado constitution called the Taxpayer Bill of Rights (TABOR). Adopted by popular initiative in 1992, it's the strictest tax-limitation measure passed by the voters in any state. Despite its name, it has nothing to do with rights of individual taxpayers and everything to do with hamstringing state government. Under TABOR, no unit of government, from the legislature to local boards, can raise taxes or approve a new tax without a vote of the people. In addition, if existing taxes bring in revenue greater than "inflation plus the percentage change in state population" for the year previous, that "surplus" must be refunded to the taxpayers. In short, TABOR froze state government in its existing shape as of 1992, and left the legislature to flounder helplessly.

In 2011, a group of taxpayers and legislatures brought suit in federal district court, arguing that a state in which no representative body has the power to tax is not a "republic" in any sense of the word.

Their quest seemed far-fetched at the time. First, citizens or even legislators who don’t like the way things are being run generally don’t have what’s called "standing to sue." Standing requires a plaintiff to show that a government action or policy—in this case TABOR—injures him or her in a way it doesn’t hurt everyone. Since TABOR affects everyone no one might have standing.

Second, the Supreme Court has for a century and a half refused to get into "republican form" disputes. In 1849, the Supreme Court refused to settle a full-scale civil war in Rhode Island by choosing one of the factions as the proper "republican" government. That was for Congress to decide, Chief Justice Roger B. Taney (later the author of Dred Scott) wrote, not for the courts. A century ago, the state of Oregon adopted the initiative system; the newly empowered citizenry promptly voted a tax on some corporate revenue. An aggrieved corporation sued, claiming that allowing government by popular vote was not "republican." No question of "standing" here, being taxed counts as "injury," but the Court held that the dispute was a "political question," beyond the jurisdiction of the courts. Chief Justice Edward White noted that the challenge was not simply "on the tax as a tax, but on the State as a State." Such a broad attack on a state was "not therefore within the reach of judicial power."