While I don't suffer from low iron, this is a problem my younger sister is continuously dealing with. I've seen how fatigued she can get, especially after her periods as you mentioned. It's scary! I can only imagine what you two must be going through! Plus she's a picky eater--no meat, hates vegetables--so she's had to work around that. With her dr's help, she's found that lentils and beans are good sources of iron, foods cooked in cast-iron pans absorb more iron, and that most breakfast cereals are fortified with iron.
Do you take your iron supplement with vitamin C? This is supposed to make you absorb the iron better, but maybe the vitamin would lessen the constipation. I've also heard of iron supplements combined with stool softeners.
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Re: Iron
With her dr's help, she's found that lentils and beans are good sources of iron, foods cooked in cast-iron pans absorb more iron, and that most breakfast cereals are fortified with iron. 
