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The only real option to bleach is iodine. Either crystals, or plain (clear, not red colored) "tincture of iodine" which is iodine dissolved in alcohol.
Bear in mind that common table salt is IODIZED because most of us do not get enough iodine in our diets. Too much iodine will leave a taste in the water, but a charcoal filter or other means will remove that. You can get iodine from most pharmacies, definitely tincture if they don't have crystals. "Polar Pure" water purifier is a special jar kit of iodine crystals.
If you wanted to do a little investing and go chemical free, UV-C ("C" band UV lighting) breaks down DNA quite aggressively. Bacteria, viruses, parasites, all get disrupted by it in no time flat. You can buy a UV-C water treatment lamp, which is normally installed as the last part of a home water purificiation system, and either install it before your faucet, or in your water tank. There are even two portable versions of this ("SteriPen" is one) sold in camping stores for treating water.