The news comes one day before a virtual news conference to be held by Alison Levitt QC, who has led an independent review into the governance scandal which engulfed RICS in recent months.
Tompkins is the third high profile departure from RICS according to Property Week.
In August, managing director Matthew Howell left to take up a position at the British Medical Association, and in June chief operating officer Violetta Parylo resigned unexpectedly.
The controversy follows allegations earlier this year in the Sunday Times about apparent failures by RICS senior figures to act on a report by accountancy firm BDO.
The newspaper allegations also claimed that four RICS non-executive directors who wanted the organisation to act on the report’s concerns had their appointments terminated.
On August 12 this year Levitt, the independent QC asked to prepare a report into the financial conduct row at RICS, delivered a 400 page report to the chair of the steering committee of the governing council.
At the time Levitt advised the members of the governing council to take time to consider her report before deciding what action to take in relation to it.