Luke,
The offspring of the homer and fantail will be a crossbreed, neither homer or fantail because it (the offspring) will not breed true to either breed. In fact, it probably will not breed true to itself; that is, its offspring, even if mated to another half homer-half fantail most likely will not breed true. By not breeding true I mean that the offspring will not necessarily resemble the parents, and there may be throwbacks to either original breed, which may look more like a homer or more like a fantail than the 50%-50% parents. In effect, what the offspring will be the equivalent to a mongrel, which results when two different breds of purbred dogs interbreed. Sometimes inter-breeding can result in a new breed, which means that the offspring of two similarly interbred parents may breed true, but this rarely, if ever, occurs as a result of one simple crossbreeding of two different types of purebreds.
Sometimes it may be beneficial to allow an interbreeding, but this does not result in a new breed because the ofspring will not breed true. It may, however be beneficial to strengthen the offspring, healthwise and even for increasing size of the offspring. I noticed that this happened from the accidental crossbreeding of a male white carneau and a female white king. The several sets of offspring from this pair were slightly larger and heavier than the other white carneaus that I had, but about the same size as the only pair of kings I had. The larger size may have resulted merely because of the mixing "new bloodlines in dirds that I suspect as suffered some denigration from too much inbreeding. The larger size also may merely have been a reflection of kings being a slightly heavier breed than carneaus (I do not know for certain whether kings are a heavier breed than carneaus).
Generally, it is not a good idea to crossbreed, unless the resulting offspring are for personal use, say for eating, or simply for the enjoyment of experimentation. When I was about 9 years old I had two male homers that mated with two female"drummers"or trumpeters. One pair had one offspring that had the coloring of the homer, feathers on the feet like the trumpeter, but no top-knot like a trumpeter. That offspring made a good meal, but it had no breeding value, and that is the case with most crossbreeds.
Sometimes a crossbreed is allowed to breed back to a purebred kind like of its parents. If this breeding back to purebreeds of that same kind continues for several generations, characteristics of the other purebred involved in the original crossbreeding may virtually disappear. Nevertheless, the bloodline of that flock of purebreds will none be genetically contaminated. Those birds carrying the genes of the purebred kind not involved in the breeding of subsequent generations may still have offspring that are throwbacks or that have characteristics different from either of the two kinds of purebreds that originally interbred. If careful records are not kept or the identity of the birds carrying the genes of the other "breed" cannot be identified, the purebred status of the entire flock has been destroyed.
It should be noted that the wild pigeons we see in our cities and fields are descended from many kinds of purebred domestic birds. As these breeds interbred the characteristics of their offspring reverted back to those of the original ancesters of the domestic purebred breeds, the "blue rock dove" of Egypt. The many breeds we have today were developed through careful selective breeding and experimentation with various combinations of inter-breeding. When the resulting offspring possessed the sought-after characteristics or some set of very desireable characteristics, some similar but some distinctly different from their forebears, and the offspring bred true, (that is,they and their offspring consistently produced offspring that resembled their parents, with their new distinctive characteristics), a new breed was established. Members of new breed were no longer merely a group of crossbreeds, but purebreds that would pass on their distinctive characteristics to their young.
Sorry for the length of this reply, but I hope it persuades you to acquire appropriate partners of the same kind for your fantail and homer.
Gordmeister