Halacha forbids eating meat together with fish, and it is considered dangerous to do so. One must also not use the same plate or the same cutlery for meat and fish. Special attention should be paid at weddings, where often fish and meat are both served. One must ensure not to eat meat and fish together, and separate plates and cutlery must be used.
It is permissible to eat meat immediately after eating fish, and indeed, many people have the custom to eat fish and then meat at the Friday night meal (with separate plates and cutlery, as mentioned). Preferably, one should wash his hands and eat something else in between the fish and the meat.
The Halachic authorities address the question of whether it is likewise permissible to eat fish immediately after meat. Meat particles occasionally become lodged in the teeth, and thus we should perhaps be concerned that if one eats fish after eating meat, some meat that had been caught in his teeth will be dislodged while he eats the fish, such that he eats meat and fish together. (Fish particles generally do not tend to become wedged in between teeth, and thus there is no such concern when one eats meat after fish.) For this reason, the Aruch Ha'shulhan (Rav Yechiel Michel Epstein of Nevarduk, 1829-1908) writes (Y.D. 173:2) that the custom in his locale was not to eat fish after eating meat. He acknowledges, however, that other communities permitted doing so. It should be noted that the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in Parashat Naso (10), writes that he followed a custom to eat some bread together with fish at the end of the Shabbat meal. Presumably, the Ben Ish Hai ate meat during the meal, and this would thus seemingly prove that he permitted eating fish after meat.
It would appear that this issue depends upon communal custom, and since there does not seem to be an accepted custom on this matter in our community, one may be lenient and eat fish after meat. It would be proper, however, to first wash one's hands and eat something else in between, as is customarily done when wishes to eat meat after fish.
(See Mehkereh Eretz, 4:6.)
Summary: One may not eat fish and meat together. One may eat one after the other, but he should preferably wash his hands and eat something else in between.

Daily Halacha
Eating Meat After Fish
Lectured by Rabbi Eli Mansour
