Your only choice, then, is to list tax-exclusive unit prices. Your decision to display tax-inclusive prices in the shop is a marketing choice. It has nothing to do with accounting.
Also, your example is incorrect. If the tax-inclusive price is 150 EUR, for 10% tax, the price without tax would be 136.36. (136.36 x 1.10 = 150.00, neglecting rounding error) Manager calculates what the tax-exclusive price would have to be to produce the tax-inclusive price you specify. It does not subtract the tax percentage from the tax-inclusive price. This is explained in the Guide linked to above. This is often difficult to explain to customers, and is one of the reasons using tax-exclusive pricing is more common, especially on expensive items. In fact, many jurisdictions prohibit tax-inclusive pricing except on inexpensive items in cash environments, such as selling refreshments at public events.