Radoh said:
Asexuality actually means that you have no sex organs and reproduce the a cloud of spores or mitosis.
Non-sexuality is what I believe you are looking for.
http://en.wikipedia.org/wiki/Asexual
I believe Asexual Reproduction is what you are looking for, as you are implying that there is reproduction happening when you refer to fungi, asexuality simply means "lack of sexuality", and since humans always talk about themselves, we can assume in this context it doesn't actually mean "lack of sexual organs", such as in amoeba, rather, your feelings towards sex.
As far as I can tell, asexuality and non-sexuality mean the same thing, just different languages' prefixes(lack of sexuality|no sexuality).