In US research groups, at least in the sciences, its fairly common to see people who have a degree in A (either bachelors or masters) and then seek PhDs in B. Many of the skills are interchangeable, and a variety of viewpoints and skillsets is important to many research problems. However, while I'd not be surprised to see someone with a Math, Computer Science, or Biology background in my Chemical Engineering lab, I'd be rather shocked to find someone with a solely English or Philosophy background. I expect that you'll run into a couple of problems, based on the gap between your background A and your target field B:
- Recruiting committees for B want to make sure that you have the skills to succeed in researching B. If you don't have a degree in B or a related field, you'll have to explain to them why your A skills prepare you for B. Lacking that connection, explain what other proof you have that you can do well in studying B. You've got research experience, but I'm not sure engineering research experience would perfectly prepare you for an English program, for example.
- There is typically a coursework component for PhDs, and those courses assume that you have an undergraduate understanding of B. If you don't, you have to delay taking the graduate level courses in B to take effectively remedial lessons. The greater the overlap, the less remedial content you need, which means you spend more time researching and less time sitting in class, which makes your department happier. All incoming PhD students need time to learn the field, but it can be hard to get started on a good foot when you need a bunch of classes to understand the basics.
When you read papers and materials in B, ponder what frustrates you about B-researchers from your perspective as an A-researcher (e.g. a lack of statistical rigor, a lack of consideration for cultural elements, too much emphasis on the results and not on elegance of the process). Some of those frustrations may fade as you become more immersed in B, but some may stay, and those frustrations are where the difference in perspective coming from A is important. Those perspectives are valuable in the B environment.