Nope, it's not LTspice's fault; LTspice is actually a pretty good program.
Consider the equivalent circuit when the diodes turn off (which is the period under question). When they turn off, they become open circuits (with some parallel capacitance). Thus, the two diodes have roughly zero current flowing through them, but substantial voltage across them. Since you have two diodes in series, the voltage they block (12V) will be divided evenly between them. And that's why you see 6V across each diode during the off period.