It's fine to have small straps under the tank. I have two racerback shelf bras, but I've had them for years, because I only wear them a few times in the summer.

Just wear a racerback bra. Or you can convert a regular bra to racerback using a special clip that pulls the straps together (google it).

Any suggestions for a bra with good support that is backless and strapless? I have a racerback t shirt with a built in shelf bra and need more support.