What would be the best way for replacing egg wash when baking bread?
I've tried unsweetened soy milk but it didn't brown that well.
You could always try using vegetable, corn, or light/regular olive oil, vegan margarine, or light corn syrup thinned with a bit of water (to prevent over browning):
I also saw something here that mentioned the use of soy milk, but you said it didn't brown well. If it didn't brown at all, then it is most likely because you were using unsweetened soy milk (remember, sugar browns when exposed to heat).
An effective method I've recently tried is using date honey, diluted with water or almond milk at a ratio of 1:1.
Another fine substitute is carob syrup, diluted similarly.
Due to the dark color of both ingredients, browning is guaranteed.
The two options also work well for browning grilled vegetables or tofu.