What do sundials tell us at night?

The answer to this question is simple: in clear weather and sufficient moonlight sundials will show local sun time. On full moon days, time is determined directly from the shadow and existing dial markings, on other days a correction, depending on the phase of the Moon is needed. In the days close to new Moon lighting is not sufficient, and sundials, alas, doesn't not work.

To know the Moon phase, you can use the following method. First, you need to determine "year's number" Y, basing on following 19-year cycles: 2014 - 1, 2015 - 2, 2016 - 3, ... 2022 - 19, 2023 - again 1. If D - day M - month, the age of moon in days L can be calculated by formula:

L = mod₃₀(11Y + D + M - 14),

where the operation mod₃₀ defines modulo 30 addition. For example, it is possible to calculate the age of the Moon on May 10, 2013 as follows:
1. The number Y for 2013 equal to 19.
2. Sum 209 (19 × 11), 10 (day) and 5 (month). Subtracting 14 from 224 (the sum) we get 210.
3. 210 Module 30 of 210 equals to 0 lunar day (210 - 7×30 = 0), which corresponds to the new moon. This is true, since there was annular solar Eclipse on this day, and it could be observed in Australia. Thus, a value of 0 will correspond to the new moon, 7th day of the moon - first quarter, 14-15th day - full moon, 22 last day of quarter.

On the day of full moon (14) sundials will show time with 12-hours correction. Or, no correction, if hour lines are marked in 12-hours format. Starting from this day, the moon begins to "lag" from the Sun for 48 minutes a day. Moon's shadow is clearly visible the week before and week after the full moon. Below is a table for those moon days, when moon lighting is suffucient.

The photo shows moondials at the astronomical site in Valencia. Using this tool you can see the times of moonrise/moonset and the time of Meridian transit for the different phases of the moon.