Sauce – Orange sauce for goose

I would think the main factor in the colouration, assuming you use the same amount of ketchup each time, is the colour of the egg yolks.

There is a specified range of yolk colours, specified on a yolk colour fan (PDF). The colour varies depending on the amount of caretonoids in the feed the chickens have eaten. Most chicken farms will supplement the feed with supplemental caretonoids in order to avoid pale yolks.

As eggs from the supermarket will likely vary widely in source, and thus feed, supplements and colour, it may be worthwhile to find a local, small-scale farm where you can ask the farmer about feed and supplements. If the chickens are truly free range, they may well eat a lot of insects, which also contributes to a rich orange yolk.

There are a number or styles of soy sauces used in Japanese cuisine. Saishikomi shoyu is sometimes described as sweet compared to regular Koikuchi shoyu. Additionally there are soy-based sauces available in the US that have added ingredients such as corn syrup or MSG to enhance specific flavors.

Kikkoman makes an "enhanced" soy sauce that they call Amakuchi shoyu that is sweetened with corn syrup, as well as another call Umukuchi shoyu that has added MSG for savoriness. Teriyaki sauces, especially those made for use as a marinade, can be as liquid as standard soy sauce and are sweetened with sugar or corn syrup.

A traditional dipping sauce for shabu shabu is a 1:1:1 mixture of shoyu, dashi, and mirin reduced by 1/3 with rice vinegar added to taste. If you wanted it even sweeter, you could just add sugar to taste once it has reduced.