Eastham rhymes with:
accustom, argosystem, biosystem, cablesystem, custom, ecosystem, subsystem, system, techsystem, telesystem, ultrasystem
Eastham sounds like:
eastin, eastman, easton, echidna, eckstein, ecosystem, ecton, eighteen, ejection, equation, equitation, estan, esteem, estonia, estonian, exaction, excitation, execution, executone, exhaustion, exton, eyestone