Generally, the pair of complex numbers that multiply to give a real number are represented by points that are in adjacent quadrants (Eg: if z1 is in the first quadrant, z2 will be in the second or the fourth one). However, this is not enough to make the product of z1 and z2 a real number. The other requirement for this to happen is that the product of the real part of z1 and imaginary part of z2 has to be equal to the negative product of the real part of z2 and the imaginary part of z. This is so because then the sum of the two products will equal to 0:
z1 = a+bi , z2 = c-di
z1z2 = ac-adi+cbi+bd
As stated above, z1z2 will only give a real number if ad = cb, because in that case, -adi + cbi = 0, leaving us with ac+bd (a real number). An example is shown in the image.
A much simpler scenario would be when both the points are on the same axis. If both points are kept on the real axis, then there is no imaginary value, thus leaving us with a real value. If both points are on the imaginary axis (real = 0) z1z2 would basically be bi * -di (since a and c = 0), this would give us bd (i*i = -1), which, again, is a real value.
Now, moving on to the next question:
The products of the complex number give an imaginary number when:
1) The two points are in diagonally opposite quadrants.
2) The products of the real parts and the imaginary part are equal but have the opposite signs. Since that would mean that those two products will add up to 0:
z1 = a+bi , z2 = -c-di
z1z2 = -ac-adi-cbi+bd
In this case, the product will give us an imaginary number if "the product of the real parts (a*-c = -ac) and the imaginary part are equal but have the opposite signs (bi * -di = bd)", so if ac = bd, bd - ac = 0, which leaves us with -i(ad+bc)
A much more simple scenario would be keeping one of the points on the real axis and the other on the imaginary axis (in the equation given above, a and d = 0). This leaves us with z1z2 = bi * -c, which is -cbi, an imaginary number.