The Utah Democratic Party, under the leadership of chairman Randy Horiuchi, has embarked on what some call an unholy alliance - an initiative petition drive with tax limitation leaders to change state tax and minimum wage law.

Horiuchi has been toying with some kind of coalition with the mainly conservative tax limitation movement for more than a year - ever since coalition leaders Merrill Cook and Greg Beesley had their much publicized split with the Republican Party.But because the Democrats opposed the three tax-limitation petitions filed by the tax protesters, the marriage was never consummated.

It may not be much of a marriage now, even though vows have apparently been made.

Horiuchi, who is considering running for an unprecedented third term as Democratic Party chairman, is being criticized by some party heavy-hitters for jumping into the coalition.

Former party chairman Pat Shea thinks it just a bad idea. Former Democratic Gov. Scott M. Matheson wonders about it also, but especially doesn't like the idea of making tax-law decisions via initiative petition instead of in the Legislature.

The unsettling in the Democratic hierarchy may well lead to a strong challenge to Horiuchi come June's party convention - maybe the end of his party leadership.

The coalition Horiuchi wants is traditional Democrats and labor union members combined with the tax protesters. The coalition is to support at least two initiative petitions: a higher Utah minimum wage and removal of the sales tax from food. More than 65,000 signatures of registered voters will be needed to place the items on the 1990 general election ballot.

Both raising the minimum wage and removing the sales tax from food, previously pushed by Democratic legislators, have found little favor in the Republican-dominated Legislature.

Going directly to the people via the petitions has a number of overtones, Shea believes.

One, it could anger the Republicans in the Legislature so much that the majority party could punish Democrats and their programs for years.

It falls upon the majority party to formulate budgets and set taxing policy. For Democrats and tax protesters to take away that treasured responsibility - one that is jealously guarded by the Republicans - could really lead to hard feelings.

Two, removing the sales tax from food completely would cost upwards of $100 million in state revenue. If that wasn't made up by increasing some other tax, (Cook and the tax protesters don't want it made up by raising some other tax), education and social programs - long the rally programs of Democrats - could actually be harmed.

Thus, Democrats could be shooting themselves in the foot by supporting a sales tax reduction on food through the initiative process.

Horiuchi sees things differently, of course.

He longs to broaden the base of the Democratic Party in Utah. And he's found it a tough task.

Even with conservative Utahns getting angry with GOP leaders for raising taxes in 1987 - and they were probably as mad as they could get these past two years - it seems that Republican leaders can always pull them back on election day by raising moral or liberal issues embraced by the national Democratic Party.

Such a coalition as Horiuchi suggests could, argue some Democrats, lead to more trouble than help - and not deliver any votes for Democrats in legislative or other state races in any case.

As one GOP leader told me after reading about the coalition: "Cook and Beesley ripped us up (when Cook, a former Republican, ran for governor as an independent). Now maybe they can do the same thing to the Democrats. Wouldn't that be great."