Provide an algorithm to multiply a series of Base 3 integers, represented as strings, without converting to any other base during any part of the algorithm. Each string (Base 3 integer) may have an arbitrary number of digits.

You may not use arithmetic operations on the digits in a way that is equivalent to changing the base, although you may use a dictionary or table lookup to recognize, for example, that in Base 3, 2*2 = 11.

You can use any programming language, or pseudo-code, or provide a verbal description.