Fish oil , especially high dose, comes  with a modest increase in bleeding time (and some incidence of bradyarrhythmias); thus many cardiologists have a concern for prescribing in patients taking oral anticoags.  Though in literature the risk appears low. 

UpToDate:

 SAFETY — In small trials, fish oil capsules up to 12 g/day (containing 6 g/day n-3 polyunsaturated fatty acids [n-3 PUFA]) have been administered for more than two years without serious adverse events [104,105].

 Although even these very high doses of fish oil appear to be safe, the US Food and Drug Administration (FDA) recommends that the general population not exceed 3 g/day of eicosapentaenoic acid (EPA) and docosahexaenoic acid (DHA) combined, with up to 2 g/day from dietary supplements, without the guidance of a clinician [106,107]. Higher dosing is approved by the FDA under the guidance of a clinician.

 The FDA has approved fish oil at a dose of 4 g/day for prescription therapy of hypertriglyceridemia [101]. In the REDUCE-IT trial, which used this formulation, overall rates of adverse events and serious adverse events were generally similar between treatment and placebo groups, with a few small differences [108]. The few differences should be interpreted cautiously given multiple comparisons and lack of prespecified hypothesis or plausible biology for such effects.

 Bleeding — In a systematic review of n-3 PUFA for the primary and secondary prevention of cardiovascular disease, high-dose supplementation did not increase the risk of bleeding (relative risk [RR] 1.06, 95% CI 0.73-1.52; eight trials, n = over 45,000) [109]. These findings are reassuring as some randomized trials of fish oil supplementation (often 6 g/day or more) have included patients at relatively high risk for bleeding, including patients undergoing percutaneous coronary intervention, carotid endarterectomy, and cardiac surgery.

 In a detailed secondary analysis of the OPERA trial, high-dose n-3 PUFA (8 to 10 g loading dose over two to five days preoperatively, followed by 2 g/day postoperatively) did not increase bleeding risk among a multinational general population of 1516 patients undergoing cardiac surgery [110]. Compared with placebo, risk of BRAC (Bleeding Academic Research Consortium) bleeding was not higher in the fish oil group (odds ratio [OR] 0.81; 95% CI, 0.53-1.24; absolute risk difference, 1.1 percent lower [95% CI, -3.0 to 1.8]) and was associated with significantly fewer total units of red blood cell transfusions (1.61 versus 1.92 units in fish oil versus placebo). These findings suggest little need to hold fish oil or delay procedures before cardiac surgery.