How to test difference between two unpaired groups, low sample numbers, many zeros?


I have a problems that at the beginning seemed easy, but now it looks trickier than expected.
I have two series of responses (well different series, but all of them with the same characteristics). In one, the values are almost always zeros. In the other quite often I have a response, like:
x1 = 0, 0, 0, 0.5, 0
y1 = 0.3, 0.2, 0.1, 0, 0.4

Now, I'm looking for a way to statistically compare them. At a first glance it seemed that Mann-Whitney test could have been used, but with that many zeros as I understand it is not reliable (because of the ties in the ranking).
I've read of "two-parts models", but I cannot find many information (and software)

I thought of dichotomizing the responses. From the experiment point of view it is perfectly plausible. But then, in which way can i test the differences?
This potentially can be the best option...

Any help will be greatly appreciated...

Thanks in advance..