I love chutneys and relishes, so am found peering at the carb count of boutique relish brands in the supermarket aisle, often. I find, as well as being the lowest carb kinds of chutneys - a little goes a long way.
Looking at recipes online, the key is sugar substitution/alternatives. And therefore, homemade is not as longlasting, I think. But maybe that's not an issue! Or, it can be frozen in batches to take out at intervals. The recipes look wonderful, in terms of yummy ingredients, and all friendly to low-carbing. Take your pick! Ditto with sugar substitutes/alternatives.