In principle, we can compute the character table of a finite group algorithmically. For example, [McKay, J. K. S. A method for computing the character table of a finite group. 1968 Computers in Mathematical Research pp. 140--148 North-Holland, Amsterdam MR0236278 (38 #4575)] and 4575)], [McKay, J. K. S. Algorithm 307. Comm. ACM 10, 7, (July 1967) 450-451.]450-451.], Dixon, John D. High speed computation of group characters. Numer. Math. 10 1967 446--450. MR0224726 (37 #325)] and others; the ideas go back to Burnside, at least, it seems. McKay's program was used to compute the characters of $J_1$ and $J_3$ in all of 84 minutes at the time, with a whole 16K store (which is 12 times smaller than the size of the background image I am using as background to my desktop computer!)
Using a bit of character theory, as in Noah's answer, we can then tell which characters are afforded by real representations, and what are the real faithful representations.
The answer to your question is thus: yes.