There was a realignment of sorts in the 1890s, but I’m not sure I would call it one of the major party realignments of U.S. history.
During Reconstruction and immediately after, the Republican and Democratic parties were largely regional parties, but with a couple of ethnic twists. The Northeast and Midwest were generally Republican, but not solidly so on account of many urban “ethnic” whites voting Democrat. Southern whites voted overwhelmingly Democrat. Before the end of Reconstruction the South was not solidly Democrat because southern blacks voted overwhelmingly Republican until they were disenfranchised.
During Reconstruction, this secnario gave the Republicans safe majorities in both Houses of Congress. After Southern blacks were disenfranchised, the South became solidly Democratic. Along with significant numbers of Northern (especially urban) Congressmen, this gave control of the House to the Democrats for 16 out of 20 years between 1875 and 1895. The greater number of Republican states in the North, Midwest, and Far West meant that the Democrats held the Senate only four of those 20 years.
Between the late 1880s and mid-1890s, economic issues–especially coinage and the tariff–began to trump regional and ethnic ones. Republicans generally supported gold coinage and little or no silver coinage. This had the effect of minimizing inflation, and in some periods led to outright deflation. Bankers and other lenders loved the Republicans for it, but poor farmers and other debtors hated them. Silver miners also despised the policy. Democrats gained seats because of it in some states dominated by agricultural and silver-mining interests.
Republicans generally favored high tariff rates, which kept out cheap foreign goods. This turned off those who spent a lot on manufactured goods (importers, farmers again, and some merchants) but tended to attract those whose livelihoods depended on manufacturing (factory owners, factory workers, steel workers, etc).
In short, during the 1890s, the Republicans lost a fair amount of support in the mining-dominated Far West and farming-dominated parts of the Midwest, but gained a lot in the industrial cities and coal- and iron-mining towns of the Midwest the Northeast. Not all urban ethnic white factory workers switched parties by any means, but enough to make a difference. It helped that silver became more or less a non-issue after about 1895-1905 when new gold discoveries in various parts of the world led willy-nilly to inflation, the objective of most silver supporters.
The realignment seems to have benefited the Republicans more than the Democrats. The Republicans held control of both Houses of Congress continuously between 1895 and 1911.
For more information, see Critical Elections and Congressional Policy Making by David W. Brady (1988), which has a chapter or two on this period.