The question really is why leave them in?
After a hysterectomy the "tubes" are redundant in fact the "post reproductive” have no known benefit in fact they may be a future health hazard.
The only argument for leaving them used to be the fear of compromising the blood supply to the ovary, however this has been shown not to be true.
Researchers comparing women who had their tubes removed and those who had them left behind after hysterectomy found no difference in hormone levels. And the timing of the menopause has not been found to be different
These are the things that can go wrong with the Fallopian tube after hysterectomy:
Infection
A number of women after hysterectomy may develop infection in the remnant fallopian tube
Hydrosalpinx where the tube becomes filled with fluid this can happen in up to 30% of women after hysterectomy however only a minority of women require reoperation for this reason
Benign cysts of the Fallopian tube:
These sometimes can grow to a very large size and can be confused with ovarian cysts. They can be responsible for pain after a hysterectomy
Ectopic pregnancy:
The first case of ectopic pregnancy after hysterectomy was reported in 1895 since then there have been 56 reports in medical literature with the number of reports on the increase. It can happen after any type of hysterectomy as long as tubes and ovaries have been left. The diagnosis of ectopic pregnancy after hysterectomy is often quite because most women and health professionals would not put ectopic pregnancy at the top of their list of diagnosis particularly after a hysterectomy. But it should be considered in every woman who has tubes and ovaries that presents with abdominal pain.
Prolapse of the Fallopian tube through the vagina
This sometimes happens after hysterectomy and can be responsible for vaginal discharge after hysterectomy
Ovarian cancer risk
There are some women who are known to be genetically at risk of ovarian cancer. These women are routinely advised to have their ovaries and tubes removed once they have completed their families.
Scientists examining the removed tubes of these women found a certain type of ovarian cancer (serous) at the end of the Fallopian tubes. Taking this research further they found that the Fallopian tube was the origin of certain types of ovarian cancer and tubal cancer even in women who did not process the gene that put them at increased risk.
For these reasons most women who are undergoing hysterectomy should ask their doctor about what will happen to their fallopian tubes