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I think the enforcement issue you raise is central to the appeal of this tax. Nothing grates more than a tax unevenly applied. Fairness above all should be a goal of the tax system. One fairness inducing aspect could mesh nicely with a cure for regressivity.

The great thing about a sales tax is that it can't be avoided through loopholes in the tax code or through 'under-the table' pay schemes. Tips, for example, are grossly under-taxed in the current system but a sales tax sees all money the same. Illegal aliens who work for cash also avoid paying for their share of public services.

The problematic regressivity of the sales-tax system could be fixed through some kind of low-income 'rebate on tax paid' system but this would only go to legal citizens and it would be based on payroll tax declaration! So tip earners who didn't pay paroll taxes on those earnings couldn't claim to have made purchases with them and would not get a paid-tax rebate for that portion of thier income. A regressive (for non-citizens) sales tax might serve as a disinsentive for both illegal immigration and payroll tax avoidance for tip earners.

Moreover, one long standing but dubious conservative argument regarding taxation is that by reducing the tax burden on the rich you encourage investment. I've never believed this effect to be as large as proposed because I'm convinced that conspicuous consumption (particularly of imported luxury goods and overseas travel) reduces this investment benefit. If you cut taxes on the rich and they buy BMWs, Mercedes and Lexuses (Lexi?), European designer clothes or take vacations in the south of France then you've only managed to stimulate foreign economies (yes this may come around but only fractionally).

However, a sales tax really does make this principle true by allowing ANYONE to cut thier own taxes (by reducing consumption) in order to save and invest! Reward frugality! Reward honesty! Encourage efficiency! Great ideas!

The problematic regressivity of the sales-tax system could be fixed through some kind of low-income 'rebate on tax paid' system but this would only go to legal citizens and it would be based on payroll tax declaration!

Don't see even a payroll declaration as a requirement at all. Just determine that minimum necessary expenditure to live on, The HSS Poverty-line is one such measure which uses price of basket of goods. Then pre-pay every legal resident the amount of sales tax that is expended on that amount (Taxrate * HSSproverty_level).

No payroll declarations, no means testing, no financial statements required at all. It treats every individual the same, and every individual pays the same tax rate at the cash register. No exceptions, no exemptions, one rate at the cash register, one amount paid per individual.